Error correcting codes (ECCs) are indispensable for reliable transmission in communication systems. Recent advancements in deep learning have catalyzed the exploration of ECC decoders based on neural networks. Among these, transformer-based neural decoders have achieved state-of-the-art decoding performance. In this paper, we propose a novel Cross-Attention Message-Passing Transformer (CrossMPT), which shares key operational principles with conventional message-passing decoders. While conventional transformer-based decoders employ a self-attention mechanism without distinguishing between magnitude and syndrome embeddings, CrossMPT updates these two types of embeddings separately and iteratively via two masked cross-attention blocks. The mask matrices are determined by the code's parity-check matrix, which explicitly captures and removes irrelevant relationships between the magnitude and syndrome embeddings. Our experimental results show that CrossMPT significantly outperforms existing neural network-based decoders for various code classes. Notably, CrossMPT achieves this decoding performance improvement while significantly reducing memory usage, computational complexity, inference time, and training time.

Ensuring extremely high reliability in channel coding is essential for 6G networks. The next-generation of ultra-reliable and low-latency communications (xURLLC) scenario within 6G networks requires frame error rate (FER) below 10-9. However, low-density parity-check (LDPC) codes, the standard in 5G new radio (NR), encounter a challenge known as the error floor phenomenon, which hinders to achieve such low frame error rates. To tackle this problem, we introduce an innovative solution: boosted neural min-sum (NMS) decoder. This decoder operates identically to conventional NMS decoders, but is trained by novel training methods including: i) boosting learning with uncorrected vectors, ii) block-wise training schedule to address the vanishing gradient issue, iii) dynamic weight sharing to minimize the number of trainable weights, iv) transfer learning to reduce the required sample count, and v) data augmentation to expedite the sampling process. Leveraging these training strategies, the boosted NMS decoder achieves the state-of-the art performance in reducing the error floor as well as superior waterfall performance. Remarkably, we fulfill the 6G xURLLC requirement for 5G LDPC codes without a severe error floor. Additionally, the boosted NMS decoder, once its weights are trained, can perform decoding without additional modules, making it highly practical for immediate application. The source code is available athttps://github.com/ghy1228/LDPC_Error_Floor.

With the broadening applications of deep learning, neural decoders have emerged as a key research focus, specifically aimed at improving the decoding performance of conventional decoding algorithms. In particular, error correction code transformer (ECCT), which utilizes the transformer architecture, has achieved state-of-the-art performance among neural network-based decoders. We present three technical contributions to significantly enhance the performance of ECCT. First, we propose a novel transformer architecture of ECCT, termed themultiple-masks ECCT (MM ECCT). We employ multiple masked self-attention blocks with different mask matrices in a parallel manner to learn diverse relationships among the codeword bits. Second, we discover that constructing mask matrices based on systematic parity check matrices (PCMs) can make the attention mapssparse, which not only enhances the decoding performance but also reduces computational complexity. Finally, we propose using complementary mask matrices derived from cyclic permutations of the systematic PCM. These complementary mask matrices are specifically designed to enhance the decoding of cyclic codes. Our extensive simulation results show that the proposed MM ECCT architecture with carefully designed mask matrices outperforms the original ECCT by a large margin, achieving state-of-the-art decoding performance among neural decoders. The source code is available at https://github.com/iil-postech/mm-ecct.

In this paper, we propose a new code design technique for GLDPC codes which we call partial doping. The proposed partial doping technique enables higher degrees of freedom in constructing protograph-based GLDPC codes. We optimize the partially-doped generalized LDPC codes over the binary erasure channels and the finite length analysis shows that it outperforms the regular LDPC codes for both code rates 1/2 and 1/4.

Low-density parity-check (LDPC) codes have been successfully commercialized in communication systems due to their strong error correction capabilities and simple decoding process. However, the error-floor phenomenon of LDPC codes, in which the error rate stops decreasing rapidly at a certain level, presents challenges for achieving extremely low error rates and deploying LDPC codes in scenarios demanding ultra-high reliability. In this work, we propose training methods for neural min-sum (NMS) decoders to eliminate the error-floor effect. First, by leveraging the boosting learning technique of ensemble networks, we divide the decoding network into two neural decoders and train the post decoder to be specialized for uncorrected words that the first decoder fails to correct. Secondly, to address the vanishing gradient issue in training, we introduce a block-wise training schedule that locally trains a block of weights while retraining the preceding block. Lastly, we show that assigning different weights to unsatisfied check nodes effectively lowers the error-floor with a minimal number of weights. By applying these training methods to standard LDPC codes, we achieve the best error-floor performance compared to other decoding methods. The proposed NMS decoder, optimized solely through novel training methods without additional modules, can be integrated into existing LDPC decoders without incurring extra hardware costs. The source code is available at https://github.com/ghy1228/LDPC_Error_Floor.

MOTIVATION: DNA-based data storage is one of the most attractive research areas for future archival storage. However, it faces the problems of high writing and reading costs for practical use. There have been many efforts to resolve this problem, but existing schemes are not fully suitable for DNA-based data storage, and more cost reduction is needed. RESULTS: We propose whole encoding and decoding procedures for DNA storage. The encoding procedure consists of a carefully designed single low-density parity-check code as an inter-oligo code, which corrects errors and dropouts efficiently. We apply new clustering and alignment methods that operate on variable-length reads to aid the decoding performance. We use edit distance and quality scores during the sequence analysis-aided decoding procedure, which can discard abnormal reads and utilize high-quality soft information. We store 548.83 KB of an image file in DNA oligos and achieve a writing cost reduction of 7.46% and a significant reading cost reduction of 26.57% and 19.41% compared with the two previous works. AVAILABILITY AND IMPLEMENTATION: Data and codes for all the algorithms proposed in this study are available at: https://github.com/sjpark0905/DNA-LDPC-codes.

Generalized low density parity check (GLDPC) codes can be constructed by replacing some single parity check (SPC) nodes in LDPC codes with generalized constraint (GC) nodes. GC nodes are defined by component codes whose minimum distance is larger than that of SPC nodes. Therefore, the variable nodes (VNs) connected to GC nodes, which are called doped VNs, are more protected than the undoped VNs. Due to this effect, we observe that the doped VNs are more robust to local cycles. The distribution of local cycles is affected by the processing VN order of the progressive edge growth (PEG) algorithm, where the latter processed (lifted) VNs tend to have more local cycles. Based on the property of doped VNs and the PEG algorithm, we show that a tangible performance gain is achieved by placing the doped VNs in the latter order of the PEG algorithm compared to the former order. The performance gain is shown with a well known GLDPC code in the literature and over both the binary erasure channel and addictive white Gaussian noise channel.

Recently, the standard ResNet-20 network was successfully implemented on the fully homomorphic encryption scheme, residue number system variant Cheon-Kim-Kim-Song (RNS-CKKS) scheme using bootstrapping, but the implementation lacks practicality due to high latency and low security level. To improve the performance, we first minimize total bootstrapping runtime using multiplexed parallel convolution that collects sparse output data for multiple channels compactly. We also propose the imaginary-removing bootstrapping to prevent the deep neural networks from catastrophic divergence during approximate ReLU operations. In addition, we optimize level consumptions and use lighter and tighter parameters. Simulation results show that we have 4.67x lower inference latency and 134x less amortized runtime (runtime per image) for ResNet-20 compared to the state-of-the-art previous work, and we achieve standard 128-bit security. Furthermore, we successfully implement ResNet-110 with high accuracy on the RNS-CKKS scheme for the first time.

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD.

The comparison operation for two numbers is one of the most frequently used operations in several applications, including deep learning. As such, lots of research has been conducted with the goal of efficiently evaluating the comparison operation in homomorphic encryption schemes. Recently, Cheonet al.(Asiacrypt 2020) proposed new comparison methods that approximated the sign function on homomorphically encrypted data using composite polynomials and proved that these methods had optimal asymptotic complexity. In this article, we propose a practically optimal method that approximates the sign function using compositions of minimax approximation polynomials. We prove that this approximation method is optimal with respect to depth consumption and the number of non-scalar multiplications. In addition, we propose a polynomial-time algorithm that determines the optimal composition of minimax approximation polynomials for the proposed homomorphic comparison operation using dynamic programming. The numerical analysis demonstrates that when minimizing runtime, the proposed comparison operation reduces the runtime by approximately 45 percent on average when compared to the previous algorithm. Likewise, when minimizing depth consumption, the proposed algorithm reduces the runtime by approximately 41 percent on average. In addition, when high precision in the comparison operation is required, the previous algorithm does not achieve 128-bit security, while the proposed algorithm does due to its small depth consumption.

MOTIVATION: In DNA storage systems, there are tradeoffs between writing and reading costs. Increasing the code rate of error-correcting codes may save writing cost, but it will need more sequence reads for data retrieval. There is potentially a way to improve sequencing and decoding processes in such a way that the reading cost induced by this tradeoff is reduced without increasing the writing cost. In past researches, clustering, alignment and decoding processes were considered as separate stages but we believe that using the information from all these processes together may improve decoding performance. Actual experiments of DNA synthesis and sequencing should be performed because simulations cannot be relied on to cover all error possibilities in practical circumstances. RESULTS: For DNA storage systems using fountain code and Reed-Solomon (RS) code, we introduce several techniques to improve the decoding performance. We designed the decoding process focusing on the cooperation of key components: Hamming-distance based clustering, discarding of abnormal sequence reads, RS error correction as well as detection and quality score-based ordering of sequences. We synthesized 513.6 KB data into DNA oligo pools and sequenced this data successfully with Illumina MiSeq instrument. Compared to Erlich's research, the proposed decoding method additionally incorporates sequence reads with minor errors which had been discarded before, and thus was able to make use of 10.6-11.9% more sequence reads from the same sequencing environment, this resulted in 6.5-8.9% reduction in the reading cost. Channel characteristics including sequence coverage and read-length distributions are provided as well. AVAILABILITY AND IMPLEMENTATION: The raw data files and the source codes of our experiments are available at: https://github.com/jhjeong0702/dna-storage.

In this paper, we propose a new construction of generalized root protograph (GRP) low-density parity-check (LDPC) codes which has a guaranteed diversity order. New hybrid-automatic request with incremental redundancy (H-ARQ-IR) is also proposed to ensure high diversity and rate-compatibility in wireless communication systems, where the GRP LDPC codes are employed along with a method called feedback encoding. The proposed H-ARQ-IR has advantages of low-complexity and near-optimal performance. In order to show its performance behavior, theoretical analysis for the proposed scheme is conducted for the block fading channel. Furthermore, numerical analysis shows that it achieves near maximum diversity order under the belief propagation (BP) decoding with binary phase shift keying (BPSK) modulation and high-order quadrature amplitude modulations (QAMs).

In research on spatially-coupled low-density parity-check (SC-LDPC) codes, rate-loss of SC-LDPC codes is one of the main issues to be addressed. One way to mitigate the rate-loss is to attach additional variable nodes with an irregular degree distribution, where the degree distribution is optimized with a constraint that the belief propagation (BP) threshold should not be degraded by attaching variable nodes. However, it is observed that the degree distribution obtained with the BP threshold constraint induces degradation of the finite-length performance. In order to address the problem, we propose new optimization methods to attach additional variable nodes while minimizing performance degradation. The proposed optimization methods are based on several design techniques including the scaling law, local threshold, expected graph evolution, differential evolution algorithms, the use of a protograph structure, and puncturing codewords. Using the optimized structure for additional variable nodes, the rate-loss of SC-LDPC codes can be reduced by more than 53% without sacrificing the finite-length performance. It is also shown that the rate-loss mitigation can be translated into a performance improvement if the proposed and the conventional SC-LDPC codes are compared at the same code rate.

Irregular quasi-cyclic (QC) low-density parity-check (LDPC) codes with the block dual-diagonal (BDD) parity structure are widely adopted in many communication standards because the BDD structure supports an efficient encoding and many degree-2 variable nodes inside are adequate for the construction of mid- to high-rate codes. However, we observe that low-rate irregular QC LDPC codes with the BDD parity structure inherently contain too many degree-2 variable nodes and suffer from error floors in high signal-to-noise ratio (SNR) region. In this paper, a generalized BDD structure including double-weight circulants as well as circulant permutation matrices is proposed for low-rate irregular QC LDPC codes with low error floors which is achieved with a little bit giving up error performance in the waterfall region. When constructing the parity part of a code with the generalized BDD structure, the portion of double-weight circulants is variable so that the resulting LDPC code can achieve a desired degree distribution including degrees 2 and 3 while supporting the efficient encoding. We show that low-rate QC LDPC codes constructed with the proposed BDD structure have better theoretical properties and lower error floor than those with the conventional BDD structure.

In this paper, we propose an interference alignment (IA) scheme for the multiple-input multiple-output (MIMO) uplink cellular network with the help of a relay which operates in half-duplex mode. The proposed scheme only requires global channel state information (CSI) knowledge at the relay, with no transmitter beamforming and time extension at the user equipment (UE), which differs from conventional IA schemes for cellular networks. We derive the feasibility condition of the proposed scheme for the general cellular network configuration and analyze the degrees-of-freedom (DoF) performance of the proposed IA scheme while providing a closed-form beamformer design at the relay. Extensions of the proposed scheme to downlink and full-duplex cellular networks are also proposed in this paper. The DoF performance of the proposed scheme is compared to those of linear IA scheme and relay-aided interference management scheme for a cellular network with no time extension. It is also shown that advantages similar to those in the uplink case can be obtained for the downlink case through the duality of a relay-aided interfering multiple-access channel (IMAC) and an interfering broadcast channel (IBC). Furthermore, the proposed scheme for a full-duplex cellular network is shown to have advantages identical to those of a number of proposed half-duplex cellular cases.

In this paper, linear index codes with multiple senders are studied, where every receiver receives encoded messages from all senders. A new fitting matrix for the multiple senders is proposed and it is proved that the minimum rank of the proposed fitting matrices is the optimal codelength of linear index codes for the multiple senders. In addition, a new type of a side information graph related with the optimal codelength is proposed and whether given side information is critical or not is studied. Furthermore, linear index codes for the cellular network scenario are studied, where each receiver can receive a subset of sub-codewords. Since some receivers cannot receive the entire codeword in the cellular network scenario, the encoding method based on the fitting matrix has to be modified. In the cellular network scenario, we propose another fitting matrix and prove that an optimal generator matrix can be found based on these fitting matrices. In addition, some properties on the optimal codelength of linear index codes for the cellular network case are studied.

In this paper, we propose new design algorithms of irregular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform degree distributions using linear programming (LP). In general, irregular SC-LDPC codes with non-uniform degree distributions are difficult to design with low complexity because their density evolution equations are multi-dimensional. To overcome this problem, proposed design algorithms are based on three main ideas: a local design of degree distributions, pre-computation of the input/output message relationship, and selection of a proper objective function. These ideas make it possible to design degree distributions of irregular SC-LDPC codes by solving low-complexity LP problems over the binary erasure channel (BEC). It is shown that the proposed irregular SC-LDPC codes designed by the proposed algorithms are superior to regular SC-LDPC codes in terms of both asymptotic and finite-length performances over the BEC. We also confirm that the proposed irregular SC-LDPC code achieves better performance compared with an optimized irregular block LDPC code in the same blocklength, which implies that the proposed design algorithms also provide a new way to construct capacity-approaching block LDPC codes.

In wireless communication systems, it is well known that inter‐symbol interference (ISI) can be avoided by using a pair of matched square‐root‐raised‐cosine (SRRC) filters in the transmitter and receiver. In an effort to find methods which minimise the number of filter taps in the matched filter, numerous studies have been done. However, in practise, when the communication specification is fixed by the standard, it is not possible to change the coefficients of the transmit filter. In this study, the authors propose a new SRRC filter design with the reduced number of filter taps for a receiver of a wireless communication system. The proposed design utilises a recursive steepest‐descent algorithm when the filter coefficients of the transmitter are fixed. That is, relaxing the ISI criterion while maintaining the stopband attenuation, the proposed receiver filter has fewer filter taps than that in the conventional case without undergoing bit error rate performance degradation. The proposed receiver filter design with the reduced number of filter taps reduces the computational complexity and the detection delay in the receiver.

We analyze the symbol measures for iterative erasure insertion and decoding of a Reed-Solomon coded SFH/MFSK system over jamming channels. In contrast to conventional erasure insertion schemes, iterative schemes do not require any preoptimized threshold or channel state information at the receiver. We confirm the performance improvement using a generalized minimum distance (GMD) decoding method with three different symbol measures. To analyze performance, we propose a new analysis framework considering the “trapped-error” probability. From analysis and the simulation results, we show that ratio-based GMD decoding has the best performance among the one-dimensional iterative erasure insertion and decoding schemes.

In wireless communication systems, the received signal power is highly fluctuated by changing of distance between transmitter and receiver, and by mobility of the receiver. Most of automatic gain controller (AGC) algorithms are developed to make the constant power of received signal. But in high adjacent channel interference (ACI) environment, the conventional algorithms do not work well. The proposed algorithm utilizes a gain splitting on variable gain amplifier (VGA) and fast Fourier transform (FFT) to enhance the receiver demodulation performance in high ACI environment. The proposed algorithm has better bit error rate (BER) performance than the conventional algorithm.

The outstanding decoding performance of spatially-coupled low-density parity-check (SC-LDPC) codes comes from wave-like propagation of reliable messages. The reliable messages are triggered by shortened (known) variable nodes in some consecutive reliable positions. However, at the cost of the improvement, shortened variable nodes cause rate-loss of SC-LDPC codes. To reduce the rate-loss, additional variable nodes (so called reliable variable nodes) can be added to the reliable positions instead of shortened variable nodes. Density evolution (DE) is an efficient method to design degree distribution of the reliable variable nodes. However, degree distributions obtained by DE show degraded performance in finite-length code performance. In this paper, we generalize the expected graph evolution and use the analysis tool in optimizing degree distribution which shows the minimum rate-loss without finite-length performance degradation. From the well-designed degree distribution, rate-loss reduction by 60% can be achieved without finite-length performance degradation.

The authors propose new McEliece cryptosystems based on punctured Reed–Muller (RM) codes. They successfully show that the commonly known attacks, such as the Minder–Shokrollahi attack, the Chizhov–Borodin attack, and the square code attack, are ineffective against the proposed RM code‐based McEliece cryptosystem. We developed an optimal puncturing scheme to prevent the above‐mentioned attacks for the proposed RM code‐based cryptosystems in a sense that the exact locations of puncturing positions with the minimum number of punctured columns of the generator matrix should be found for attacking. It is important to carry out the minimum number of punctures, however, as code modification resulting from puncturing can reduce security. Additionally, the square code attack can be prevented in the proposed RM code‐based McEliece cryptosystems by using both the proposed puncturing method and random insertion methods.

In this study, the authors propose a new two‐stage decoding scheme for low‐density parity‐check codes to lower the error‐floor, which consists of the conventional belief propagation (BP) decoding algorithm as the first‐stage decoding and the re‐decodings with manipulated log‐likelihood ratios (LLRs) of variable nodes as the second‐stage decoding. In the first‐stage decoding, an early stopping criterion is proposed for early detection of decoding failure and the candidate set of the variable nodes are determined, which can be partly included in the small trapping sets. In the second‐stage decoding, the scores of the variable nodes in the candidate set are computed by the proposed unreliable path search algorithm and the variable nodes are sorted in ascending order by their scores for the re‐decoding trials. Each re‐decoding trial is performed by BP decoding algorithm with manipulated LLR of a selected variable node in the candidate set one at a time with the second early stopping criterion. The parallel unreliable path search algorithm is also proposed for practical application of the proposed algorithm. Numerical results show that the proposed early stopping criteria and the proposed decoding algorithms for the second‐stage decoding can correct most of the unsuccessfully decoded codewords by the first‐stage decoding in the error‐floor region.

In this paper, two classes of fountain codes, called batched zigzag fountain codes and two-phase batched zigzag fountain codes, are proposed for the symbol erasure channel. At a cost of slightly lengthened code symbols, the involved message symbols in each batch of the proposed codes can be recovered by low complexity zigzag decoding algorithm. Thus, the proposed codes have low buffer occupancy during decoding process. These features are suitable for broadcasting to receivers with limited hardware resources. We also propose a method to obtain degree distributions of code symbols for the proposed codes via ripple size evolution by taking into account the released code symbols from the batches. We also show that the proposed codes outperform Luby transform codes and zigzag decodable fountain codes with respect to intermediate recovery rate and coding overhead when message length is short, symbol erasure rate is low, and available buffer size is limited.

In this paper, new index coding problems are studied, where each receiver has erroneous side information. Although side information is a crucial part of index coding, the existence of erroneous side information has not been considered yet. We study an index code with receivers that have erroneous side information symbols in the error-free broadcast channel, which is called an index code with side information errors (ICSIE). The encoding and decoding procedures of the ICSIE are proposed, based on the syndrome decoding. Then, we derive the bounds on the optimal codelength of the proposed index code with erroneous side information. Furthermore, we introduce a special graph for the proposed index coding problem, called a δs-cycle whose properties are similar to those of the cycle in the conventional index coding problem. Properties of the ICSIE are also discussed in the δs-cycle and clique. Finally, the proposed ICSIE is generalized to an index code for the scenario having both additive channel errors and side information errors, called a generalized error correcting index code.

Partial transmit sequence (PTS), a well‐known peak‐to‐average power ratio (PAPR) reduction scheme for orthogonal frequency division multiplexing (OFDM) systems, has been actively investigated to reduce its high computational complexity. Ku et al . proposed a selection method of dominant time‐domain samples and by only using the selected samples, the PAPR of each alternative OFDM signal vector is calculated. This method clearly reduces the computational complexity but it is crucial to select proper time‐domain samples to achieve acceptable PAPR reduction performance. In this study, a new selection method of dominant time‐domain samples is proposed based on rotating samples of inverse fast Fourier transformed (IFFTed) subblocks to the local area on which the corresponding sample of the IFFTed first subblock is located. Moreover, pre‐exclusion of phase rotating vectors based on the above time‐domain sample rotation is proposed to further reduce the computational complexity. Numerical results confirm that the proposed PTS schemes substantially reduce the computational complexity with negligible degradation of PAPR reduction performance.

As a metric for amplitude fluctuation of orthogonal frequency division multiplexing (OFDM) signal, cubic metric (CM) has received an increasing attention because it is more closely related to the distortion induced by nonlinear devices than the well-known peak-to-average power ratio (PAPR). In this letter, the properties of CM of OFDM signal is investigated. First, asymptotic distribution of CM is derived. Second, it is verified that 1.7 times oversampling rate is good enough to capture the CM of continuous OFDM signals in terms of mean square error, which is also practically meaningful because the fast Fourier transform size is typically 1.7 times larger than the nominal bandwidth in the long-term evolution (LTE) cellular communication systems.

An interference alignment (IA) scheme for interference channels was recently proposed to achieve the maximum degrees of freedom (DoF). Although most studies of IA focus on network throughput, that is, DoF, reliability in terms of diversity order is also an important performance measure. In this study, interference cancellation (IC) scheme based on Alamouti code in the multi‐access scenario is applied to the K ‐user, multi‐input–multi‐output (MIMO) interference channel. This IC scheme gives the benefit of diversity order and requires no channel‐state‐information at the transmitters (CSIT). However, it requires more receive antennas than the IA scheme to achieve the same DoF. In order to reduce the number of receive antennas, especially for the three‐user MIMO interference channel, an IA‐and‐cancellation (IAC) scheme based on Alamouti code is proposed. It keeps the same DoF as the IC scheme, but requires partial CSIT. It is analytically shown that the IC and IAC schemes enable symbol‐by‐symbol decoding and achieve diversity order of two, while the conventional IA scheme achieves diversity order of one.

In this study, the minimum number of antennas at each user is derived to obtain 2 M 2 degrees of freedom (DOF) in the M × 1 × M multiple‐input–multiple‐output (MIMO) multi‐user two‐way relay X channels. Based on the design of beamforming vectors for the conventional signal space alignment scheme, it is shown that each user needs at least ⌈ M 2 − ( M /2)⌉ antennas to obtain DOF 2 M 2 with the relay having M 2 antennas. As the number of users increases, the number of antennas also increases, which makes the implementation difficult. In an effort to reduce the number of antennas, the authors propose a new beamforming scheme of MIMO multi‐user two‐way relay X channels using the time extension. Through the numerical analysis, it is confirmed that the proposed scheme with time extension is a good alternative scheme to replace the conventional scheme.

Let p be an odd prime and n = 2m with pm≡ 1 (mod 4). In this paper, the cross-correlation distribution between two decimated sequences of a p-ary m-sequence, s(2t + i) with i ∈ {0, 1} and s(d't) with d' = 2d, d = (pm+1/2)2is determined.

In this study, two relay‐aided interference alignment (IA) schemes are proposed for the quasi‐static M × 2 X channel where M is the number of transmitters and two is the number of receivers, respectively. The first scheme utilises one full‐duplex relay and the second one utilises two half‐duplex relays. In the proposed schemes, the transmitters transmit signals in every time slot and the relays operate in the amplify‐and‐forward (AF) mode. By verifying the linear independence between the desired signals and the interference, it is shown that the proposed schemes can achieve 2 M /( M + 1) degrees of freedom (DoF) which is the maximum DoF for the M × 2 X channel. It is also shown by using the concept of reciprocity that the proposed IA schemes are applicable to the 2 × M X channel. Finally, through the numerical analysis, it is confirmed that the proposed IA schemes provide good alternatives to the previously known relay‐aided IA scheme.

In this paper, a new class of (N, K) near-optimal partial Hadamard codebooks is proposed. The construction of the proposed codebooks from Hadamard matrices is based on binary row selection sequences, which are generated by quadratic have parameters N = pnand K = (p - 1/2 p)(N + √N) + 1 for an odd prime p and an even positive integer n. We prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the Welch bound equality for sufficiently large p and derive their inner product distribution.

In this study, a new peak‐to‐average power ratio (PAPR) reduction scheme for orthogonal frequency division multiplexing (OFDM) is proposed based on the selected mapping (SLM) scheme. The proposed SLM scheme generates alternative OFDM signal sequences by cyclically shifting the connections in each subblock at an intermediate stage of inverse fast Fourier transform (IFFT). Compared with the conventional SLM scheme, the proposed SLM scheme achieves similar PAPR reduction performance with much lower computational complexity and no bit error rate degradation. The performance of the proposed SLM scheme is analysed mathematically and verified through numerical analysis. Also, it is shown that the proposed SLM scheme has the lowest computational complexity among the existing low‐complexity SLM schemes exploiting the signals at an intermediate stage of IFFT.

For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to have short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, based on special classes of cyclic difference families, we propose a new construction method of high-rate regular QC LDPC codes having parity-check matrices consisting of a single row of circulants with column-weight 3 or 4. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for given column-weight and design rate under girth 6. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank for column-weight 3 and just one redundant row for column-weight 4. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.

There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them focus on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple (i.e., parallel) edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on a graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and another method is proposed for regular QC LDPC codes with girth at least 14. Moreover, a construction algorithm of QC LDPC codes based on certain liftings of multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes with progressive edge-growth (PEG) LDPC codes and with PEG QC LDPC codes.

In this paper, a new combining scheme of interference alignment and Alamouti codes is proposed for the quasistatic MIMO K × 2 X channel. In fact, a brilliant scheme was already proposed in [1] for 2×2 MIMO X channel. However, it is difficult to extend for more general case. Therefore, for K × 2 MIMO X channel, a simple transmission and decoding schemes is proposed in this paper. The transmission scheme is based on the interference alignment concept and Alamouti-code and the decoding scheme exploits the zero-forcing and decoupling method of two Alamouti-codes.

In this paper, for an odd prime p such that p ≡ 3 mod 4, odd n, and d = (pn+ 1)/(pk+ 1) + (pn- l)/2 with k|n, the value distribution of the exponential sum S(a, b) when a and b run through Fpn is calculated. The weight distribution of the relevant cyclic code C over Fpwith the length L = pn- 1 and the dimension dimFpC = 2n In is also derived. Our result generalizes the case in [5].

In this paper, for an integer k, we evaluate an upper bound for the cross-correlation of a ternary m-sequence of period N = 34k+2- 1 and its decimated sequence with decimation d = 34k+2- 32k+1+2/4 + 32k+1. It is found that the cross-correlation is upper bounded by 4.5 · 32k+1+ 1.

The block dual-diagonal (BDD) parity structure is widely adopted in many practical irregular quasi-cyclic (QC) low-density parity-check (LDPC) codes. These QC LDPC codes have good error-correcting performance in waterfall region but usually show relatively high error floors in low error rate region. In this paper, by using multi-weight circulants, a new BDD structure is proposed for the parity part of irregular QC LDPC codes to lower error floors and support efficient encoding. Since the parity part of parity-check matrices has flexible degree distribution with the aid of multi-weight circulants, QC LDPC codes with the proposed BDD structure can have large minimum Hamming distance compared to those with the conventional BDD structure, especially, in the low-rate case. Simulation results show that QC LDPC codes with the proposed BDD structure have lower error floor than those with the conventional BDD structure.

In this letter, a new performance measure for compressed sensing matrices is proposed. This new measure is based on thek-set correlation vectors whose components consist of the correlation values between two columns in thek-column submatrices of a sensing matrix. This measure is highly related to the restricted isometry property (RIP). And the proposed measure has less computational complexity than the condition number approach which is a typical approach for performance prediction with RIP check. It is shown by simulation that the proposed scheme works well as a performance measure for the compressed sensing matrices.

In this paper, for an odd prime , we investigate into the cross-correlation of a p-ary m-sequence m(t) of period p;n;-1 and its d-decimated sequences m(dt+l), 0≤l;m;+1)/2, where d=(pm+1)2/2(p+l), n=2m, and m is an odd integer. There are (pm+1)/2 distinct decimated sequences m(dt+l) since gcd(d,pn-1)=(pm+1)/2. It is shown that the magnitude of the cross-correlation values is upper bounded by (p+1)/2 pn/2+1 . We also construct the sequence family F from these sequences, where the family size is pmand the correlation magnitude is upper bounded by (pm+1)/2 pn/2+1.

For a prime p ≡ 1 mod 4, an odd integer n, and d = pn+1/p+1 + pn-1/2, we investigate the cross-correlation values of p-ary m-sequence m(t) of period pn- 1 and its decimated m-sequence m(dt). It is shown that the cross-correlation function between m(t) and m(dt) takes the values in {-1, -1 ± pn/2, -1 ± 1+√p/2 pn/2, -1 ± p-1/2 pn/2}.

In this paper, two relay-aided interference alignment (IA) schemes are proposed for the quasi-static M×2 X channel. The first scheme utilizes one full-duplex relay and the second scheme utilizes two half-duplex relays. In the proposed schemes, transmitters transmit in every time slot and relays operate in amplify-and-forward (AF) mode. By confirming the linear independence between the desired signals and interference and also among the desired signals, it is shown that the proposed schemes can achieve 2M/(M + 1) degrees of freedom (DoF) which is the maximum DoF of M × 2 X channel.

In this paper, all subgraph patterns of protographs which prevent quasi-cyclic (QC) low-density parity-check (LDPC) codes from having large girth are searched in allowance with multiple edges based on graph theoretic approach. A systematic construction of protographs with multiple edges using combinatorial design is proposed for designing QC LDPC codes with girth greater than or equal to 14.

In this paper, a new family ofM-ary sequences of periodpn-1 is proposed. The proposed family is constructed by the addition of cyclic shifts of anM-ary Sidel'nikov sequence and its reverse sequence. The number of sequences contained in this family is about (M-1)2times of their period and the maximum magnitude of their correlation values is upper bounded by 4 √(pn)+5.

For an odd primepcongruent to 3 modulo 4 and an odd integern, a new family ofp-ary sequences of periodN=(pn-1)/2 with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and 2d,d=N-pn-1. The upper bound for the maximum magnitude of nontrivial correlations of this family is derived using well known Kloosterman sums. The upper bound is shown to be 2√(N+1/2) = √(2pn) , which is twice the Welch's lower bound and approximately 1.5 times the Sidelnikov's lower bound. The size of the family is 2(pn-1) , which is four times the period of sequences.

In this paper, a new relay selection scheme for decode-and-forward (DF) relay cooperative network with multiple antennas is proposed based on both channel state information (CSI) and transmission scheme by deriving the upper bound on the pairwise error probability (PEP) of the near-maximum-likelihood (near-ML) decoder. It is also proved that the proposed relay selection which selects m (1≤ m≤ N) relays from N relays achieves full diversity MSMD+N MRmin[MS, MD] regardless of the value of m in the DF relay network consisting of one source, one destination, and N relays with MS, MD, and MRantennas, respectively. Through Monte Carlo simulation, the error correction performance of the proposed relay selection for various m is shown for the uncoded single-antenna, Alamouti coded, and multiple-input multiple-output (MIMO) DF relay networks.

In this paper, we investigate into the cross-correlation of a ternary m-sequence m(t) of period 3n- 1 and its decimated sequence m(dt) by d = (3m+1)2/8, where n = 2m = 4k + 2. It is shown that the magnitude of the cross-correlation values is upper bounded by 2√(3n+ 1).

In this paper, we propose a method to construct p-ary sequence family S which has the period of N = (pn- 1)/2 when p is an odd prime and n is an odd integer. The family has the size of 2(pn- 1) and the magnitudes of correlation values between any two sequences from the family are upper bounded by 2√N + 1/2. The bound can be derived from the well-known Kloosterman sum.

In this paper, a new SLM scheme for peak to average power ratio (PAPR) reduction in orthogonal frequency division multiplexing (OFDM) systems is proposed. This scheme is special case of selected mapping (SLM) scheme. The proposed SLM scheme generates alternative OFDM signal sequences by exploiting the intermediate OFDM signal sequence in inverse fast Fourier transform (IFFT). By using this technique, the proposed SLM scheme achieves similar PAPR reduction performance with much lower computational complexity and no bit error rate (BER) degradation. the performance of the proposed SLM scheme is verified through the simulations.

In this paper, we propose the best relay selection scheme for the soft-decision-and-forward cooperative network with multiple relays. The term `best relay selection' implies that the relay having the largest end-to-end signal-to-noise ratio is selected to transmit in the second phase transmission. The approximate performances are analyzed in terms of pairwise error probability. Using the Fox's H-function, it is shown that the proposed scheme has full diversity order.

The peak to average power ratio (PAPR) reduction performance depends on the phase vectors of selected mapping (SLM) scheme where the alternative input symbol vectors are generated by multiplying an input symbol vector by phase vectors. Thus, the symbol powers of alternative orthogonal frequency division multiplexing (OFDM) signals obtained by inverse Fourier transforming the alternative input symbol vectors become correlated. In this paper, the relationship between the correlations of component powers of alternative OFDM signal vector and the correlations of phase vectors are evaluated. Then, the complementary cumulative distribution function (CCDF) of PAPR in SLM scheme is derived using multivariate gamma distribution. The derived CCDF coincides exactly with the simulation result for peak reduced OFDM system with SLM.

In this paper, a new generation method of quaternary sequences of period 2(2n-1) with ideal autocorrelation and balance property is proposed using the binary sequences of period 2n- 1 with ideal autocorrelation and reverse Gray mapping. The autocorrelation distribution of the proposed quaternary sequences is also derived.

In this paper, a near maximum-likelihood (ML) decoder for orthogonal decode-and-forward (ODF) protocol with multiple antennas is proposed and the upper bound on the average pairwise error probability (PEP) for large signal to noise ratio is also derived, regardless of the modulation scheme. From the upper bound on the average PEP, we show that the ODF protocol with near ML decoder can achieve full diversity MSMD+MRmin(MS, MD) for a cooperative network consisting of one source, one relay, and one destination with MS, MR, and MDantennas, respectively.

In this paper, for an odd prime p, new quaternary sequences of even period 2p with ideal autocorrelation property are constructed using the Legendre sequences of period p. The distribution of autocorrelation function of the proposed quaternary sequences is also derived.

We propose a new construction of quaternary sequences using the reverse Gray mapping of a pair of binary Sidel'nikov sequences. The proposed construction provides sequences of even period N with the maximum nontrivial auto-correlation magnitude, Rmax= 2. For N ≡ 0 mod 4, the new quaternary sequences have the optimal Rmax= 2 and are almost-balanced in contrast to the only earlier optimal construction Sj[1].

A cooperative protocol called soft-decision-and-forward (SDF) is introduced. SDF protocol exploits the soft decision values of the received signal at the relay node. Alamouti code is used for orthogonal transmission and distributed space-time codes are designed for non-orthogonal transmission. The maximum likelihood decoders with low decoding complexity are proposed. From simulations, it can be seen that SDF protocol outperforms AF protocol.

In this paper, the Alamouti code combined with partial response signaling (PRS) is proposed. The bit error rate of the proposed quadrature PRS (QPRS) Alamouti code is derived for the quasi-static Rayleigh fading and additive white Gaussian noise channel. Furthermore, its performance in a severely bandlimited channel is simulated by using chopping filter and compared with the conventional Alamouti code. The numerical results show that the proposed QPRS Alamouti code outperforms the conventional Alamouti code under the chopping environment.

In this paper, a new criterion for reordering of low-density parity-check (LDPC) coded orthogonal frequency division multiplexing (OFDM) subframes is proposed for type I hybrid automatic repeat request (H-ARQ) systems. It is verified from numerical analysis that a subframe reordering pattern having larger channel capacity shows better bit error rate (BER). Also, it is shown that the subframe reordering pattern achieving equal combined power allocation for each subframe maximizes the channel capacity and outperforms other subframe reordering patterns in terms of BER performance. Simulation results are provided to confirm that for a very slow varying fading channel, the proposed subframe reordering scheme for achieving equal combined power allocation gives better BER performance than the conventional Chase combining scheme without increasing the decoding complexity.

In this paper, we propose a new selected mapping (SLM) scheme for reducing peak to average power ratio (PAPR) of orthogonal frequency division multiplexing (OFDM) signals modulated with quadrature amplitude modulation (QAM), called partial bit inverted SLM (PBISLM). The proposed scheme changes the magnitudes as well as the phases of QAM symbols by applying binary phase sequences to the binary data sequence before mapped to QAM symbols. Simulation results show that the proposed scheme has better PAPR reduction performance than the conventional SLM scheme for the QAM modulated OFDM signals, especially for the small number of subcarriers.

In this paper, we propose a new blind SLM scheme with low complexity using m-sequence as a phase sequence in OFDM. The proposed scheme significantly reduces the computational complexity for searching the side information of the phase sequence and decoding the alternative symbol sequence at the receiver. After generating alternative symbol sequences using a set of phase sequences, the side information for each alternative symbol sequence is embedded in it through block partitioning and phase rotation. The proposed method does not need additional inverse fast Fourier transform (IFFT) and has the same PAPR reduction performance compared to the conventional SLM scheme. In order to find the side information, a maximum likelihood (ML) decoder with lower complexity is derived, which guarantees lower detection failure probability of side information compared to the conventional blind SLM scheme.

In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (BMMSE). By using this result, the relationship between the mutual information and the BEP is derived for multiple-input multiple-output (MIMO) communication systems with the bit-linear linear-dispersion (BLLD) codes for the Gaussian channel. From the relationship, the lower and upper bounds on the mutual information can be derived.

In this letter, the Alamouti code combined with partial response signaling (PRS) is proposed. The bit error rate of the proposed quadrature PRS (QPRS) Alamouti code is derived for the quasi-static Rayleigh fading and additive white Gaussian noise channel. Furthermore, its performance in a severely bandlimited channel is simulated by using chopping filter and compared with the conventional Alamouti code. The numerical results show that the proposed QPRS Alamouti code outperforms the conventional Alamouti code under the chopping environment.

In this paper, a new extending method of q-ary low correlation zone(LCZ) sequence sets is proposed, which is a generalization of binary LCZ sequence set by Kim, Jang, No, and Chung. Using this method, q-ary LCZ sequence set with parameters (N,M,L, isin) is extended as a q-ary LCZ sequence set with parameters (pN, pM, plfloor(L + 1)/plfloor - 1, pisin), where p is prime and p|q.

In this paper, we proposed a new quaternary low correlation zone(LCZ) sequence set with parameters (2(2n− 1),M, L, 2). The new LCZ sequence set is constructed from the binary sequence with ideal autocorrelation of period 2n− 1. The proposed construction method corresponds to the generalization of the construction method of binary LCZ sequence set by using binary sequence with ideal autocorrelation proposed by Kim, Jang, No, and Chung [1].

In this paper, for a positive integer M and a prime p such that M|pn- 1, a family of M-ary sequences using the M-ary Sidel'nikov sequences with period pn-1 is constructed. This family has its maximum magnitude of correlation values upper bounded by 3radic(pn) + 6 and the family size is (M-I)2(2n-1-1) + M - 1 for p = 2 or (M-1)2(pn-3)/2+ M (M-1)/2 for an odd prime p.

In this paper, we analyze the sequential message- passing decoding algorithm of low-density parity-check (LDPC) codes by partitioning check nodes. This decoding algorithm shows better bit error rate (BER) performance than the conventional message-passing decoding algorithm, especially for the small number of iterations. Analytical results indicate that as the number of partitioned subsets of check nodes increases, the BER performance is improved. We also derive the recursive equations for mean values of messages at check and variable nodes by using density evolution with a Gaussian approximation. From these equations, the mean values are obtained at each iteration of the sequential decoding algorithm and the corresponding BER values are calculated. They show that the sequential decoding algorithm converges faster than the conventional one. Finally, the analytical results are confirmed by the simulation results.

In this correspondence, for a positive integerMand a prime p such that M|pn- 1, three families of M-ary sequences using theM-ary Sidel'nikov sequences with period pn- 1 are constructed. Two small families contain [(pn- 1)/2]+M-2 or pn+M-3M-ary sequences, and both of their maximum magnitudes of correlation values are upper bounded by 2 radic{pn} + 6. The largest family has its maximum magnitude of correlation values upper bounded by 3 radic{pn}+5 and the family size is (M-1)2(2n-1-1) +M-1 for p = 2 or (M-1)2(pn- 3)/2 +M(M-1)/2 for an odd prime p.

For an odd prime p, n=4k, and d=((p2k+1)/2)2, there are (p2k+1)/2 distinct decimated sequences s(dt+l), 0lesl2k+)/2,of a p-ary m-sequence s(t) of period pn-1 because gcd(d,pn-1)=(p2k+1)/2. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+l),0lesl2k+1)/2, takes the values in {-1, -1 -radic(pn),-1+radic(pn),-1+2radic(pn)} and their cross-correlation distribution is also derived.

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i,i= 6,7,8,9,10, i.e., the cycles existing regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g,gges 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14.

For an odd prime p, an even integer n, and d = pk+ 1 with gcd(n, k) = 1, there are p + 1 distinct decimated sequences s(dt + l), 0 les In- 1 since gcd(d, pn-1) = p + 1. In this paper, the cross-correlation distribution between a p-ary m-sequence s(t) and its p+1 distinct decimated sequences s(dt+l) is derived. The maximum magnitude of their cross-correlation values is l+p radic pnif I = 0 mod p + 1 for n = 0 mod 4 or I = (p + l)/2 mod p + 1 for n = 2 mod 4 and otherwise, 1 + radicpn. Also by using s(t) and s(dt + I), a new family of p-ary sequences of period pn-1 is constructed, whose family size is pnmiddot and Cmaxis 1+ pradicpn.

For an odd prime p, n=4k, and d=((p2k+1)/2)2, there are (p2k+1)/2distinct decimated sequences s(dt+l), 0lesl2k+1)/2, of a p-ary m-sequence s(t) of period pn-1 because gcd(d,pn-1)=(p2k+1)/2. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+I), 0lesl2k+1)/2, takes the values in {-1, -1 - radicpn, -1+radicpn-1+2radicpn} and their cross-correlation distribution is also derived.

In this paper, we propose a new peak-to-average power ratio (PAPR) reduction scheme of orthogonal frequency division multiplexing (OFDM) system, called a modified selected mapping (SLM) scheme, which considerably reduces the computational complexity with keeping the similar PAPR reduction performance compared with the conventional SLM scheme. The proposed scheme is analytically and numerically evaluated for the OFDM system specified in the IEEE 802.16 standard. For the OFDM system with 2048 subcarriers, the proposed scheme with 4 binary phase sequences can reduce the complex multiplications by 63.5% with the similar PAPR reduction compared with the SLM scheme with 16 binary phase sequences.

In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (MMSE), that is, the BEP is greater than a quarter of the bit MMSE and less than a half of the bit MMSE. By using this result, the lower and upper bounds of the derivative of the mutual information are derived from the BEP in the multiple-input multiple-output (MIMO) communication systems with the bit-linear linear- dispersion (BLLD) codes in the Gaussian channel.

In the tone reservation (TR) scheme, it is known that the set of randomly selected peak reduction tones (PRT's) performs better than the contiguous PRT set and the interleaved PRT set in the PAPR reduction of orthogonal frequency division multiplexing (OFDM) signals. It is also known that finding the optimal PRT set is equivalent to solving the secondary peak minimization problem in the TR scheme. However, this problem cannot be solved for the practical number of tones because choosing k tones out of N subcarriers requires O(k) computational complexity. In this paper, the near optimal PRT set of the TR scheme is proposed, which is constructed from the cyclic difference set.

In this paper, we propose multicode multiple-input multiple- output (MIMO) systems with quaternary low correlation zone (LCZ) and zero correlation zone (ZCZ) sequences as spreading codes. Quaternary LCZ and ZCZ sequences have very low correlation values when the time shifts of correlation function are within the predetermined correlation zone and thus the multi-user or multipath interference can be substantially reduced when the delay is within a few chips. The bit error probability of the proposed systems is theoretically analyzed, which is numerically confirmed. It is also numerically shown that the performance of multicode MIMO systems with quaternary LCZ and ZCZ sequences is better than that of the conventional multicode MIMO systems with quaternary spreading codes constructed from a pair of binary Hadamard codes.

In this correspondence, three methods of constructing low-correlation zone (LCZ) sequences are proposed. In the first method, we constructed binary LCZ sequence sets of period 2n-1 using the Legendre sequences of period 2m-1 as a column sequence when m|n. In the second method, we devise a column sequence set of length 2m+1-1 from a binary sequence of period 2m-1 having ideal autocorrelation property and this column sequence set is used to construct binary LCZ sequence sets of period 2n-1 when (m+1)|n. In the third method, p-ary LCZ sequence sets are constructed by adopting p-ary sequence of period pm-1 with ideal autocorrelation for integers n and m such that m|n as a column sequence. The second and third methods give us the optimal sets with respect to the bound by Tang, Fan, and Matsufuji. Finally, a construction method of pntimespnp-ary Hadamard matrices from optimal LCZ sequence sets is proposed

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.

In this paper, we propose the new construction methods of constructing optimal low correlation zone (LCZ) sequences. We construct the new p-ary LCZ sequence sets by adopting p-ary sequence of period pm- 1 with ideal autocorrelation for integers n and m such that m n as a column sequence. The new construction methods give us the optimal sets with respect to the bound by Tang, Fan, and Matsufuji

In this paper, given an integer e and n such that e|n, and a prime p, we propose a method of constructing optimal p2-ary low correlation zone (LCZ) sequence set with parameters (pn- 1,pe- 1, (pn- 1)/(p - 1), 1) from a p-ary sequence of the same length with ideal autocorrelation. The resulting p2-ary LCZ sequence set can be viewed as the generalization of the optimal quaternary LCZ sequence set by Kim, Jang, No, and Chung in respect of the alphabet size. But the method used in the proof is quite different from that used in the quaternary LCZ sequence. The proof used in this paper can be used for the proof of quaternary LCZ sequence

In this paper, we define quasi-cyclic (QC) generalized Hadamard matrices and balanced QC generalized Hadamard matrices. Then we propose a new construction method for QC generalized Hadamard matrices. The proposed matrices are constructed from the balanced optimal low correlation zone (LCZ) sequence set which has correlation value -1 within low correlation zone

In this paper, we present several construction methods for low-correlation zone (LCZ) sequence sets. First, we propose a design scheme for binary LCZ sequence sets with parameters (2n+1-2,M,L,2). In this scheme, we can freely set the LCZ length L and the resulting LCZ sequence sets have the size M, which is almost optimal with respect to Tang, Fan, and Matsufuji bound. Second, given a q-ary LCZ sequence set with parameters (N,M,L,epsi) and even q, we construct another q-ary LCZ sequence set with parameters (2N,2M,L,2epsi) or (2N,2M,L-1,2epsi). Especially, the new set with parameters (2N,2M,L,2) can be optimal in terms of the set size if a q-ary optimal LCZ sequence set with parameters (N,M,L,1) is used

In this correspondence, the cycles of Tanner (3,5) quasi-cyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4,6,8, and 10 in Tanner (3,5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15th root of unity of the prime field F/sub p/. By checking the existence of solutions for these equations over F/sub p/, the girths of Tanner (3,5) QC LDPC codes are derived.

In this paper, we derive the cyclotomic numbers of order 5 over an extension field Fpnusing the well-known results of quintic Jacobi sums over Fp(B. C. Berndt, et al., 1998). For p ne 1 mod 5, we have obtained the simple closed-form expression of the cyclotomic numbers of order 5 over Fpn. For p equiv 1 mod 5, we express the cyclotomic number of order 5 over Fpnin terms of the solution of the diophantine system which is required to evaluate the cyclotomic number of order 5 over Fpn. Using the cyclotomic numbers of order 5 over Fpn, autocorrelation distributions of 5-ary Sidel'nikov sequences of period pn- 1 are also derived

In this paper, for integers e and n such that e|n and 2e- 1 is a prime, we propose a method of constructing binary low correlation zone (LCZ) sequences of period 2n- 1 by using the extended form sequence with the same period. These new LCZ sequences use Legendre sequences as their column sequences

In this paper, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel'nikov sequences. The frequency of each autocorrelation value of an M-ary Sidel'nikov sequence is expressed in terms of the cyclotomic numbers of order M. It is also pointed out that the total number of distinct autocorrelation values is dependent not only on M but also on the period of the sequence, but always less than or equal to (2M) + 1

In this paper, we derive linear complexity over Fpof the M-ary Sidel'nikov sequences using discrete Fourier transform. As an example, we represent the linear complexity of the ternary Sidel'nikov sequences. It turned out that the ternary Sidel'nikov sequences have the linear complexity nearly close to their periods

In this paper, given a composite integer n, we propose a method of constructing quaternary low correlation zone (LCZ) sequences of period 2n$1 from binary sequences of the same length with ideal autocorrelation. These new sequences are optimal with respect to the bound by Tang, Fan, and Matsufuji. The correlation distributions of these new quaternary LCZ sequences constructed from m-sequences and GMW sequences are derived

In this paper, the cycles of Tanner's (3,5) quasicyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4, 6, 8, and 10 in Tanner's (3,5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15-th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner's (3,5) QC LDPC codes are derived

In this paper, we introduce a new partial transmit sequence (PTS) orthogonal frequency division multiplexing (OFDM) scheme with low computational complexity. In the proposed scheme, 2/sup n/ - point inverse fast Fourier transform (IFFT) is divided into two parts. An input symbol sequence is partially transformed using the first l stages of IFFT to generate an intermediate signal sequence and the intermediate signal sequence is partitioned into a number of intermediate signal subsequences. Then, the remaining n - l stages of IFFT are applied to each of the intermediate signal subsequences and the resulting signal subsequences are summed after being multiplied by each member of a set of W rotating vectors to yield W distinct OFDM signal sequences. The one with the lowest peak to average power ratio (PAPR) among these OFDM signal sequences is selected for transmission. The new PTS OFDM scheme reduces the computational complexity while it shows almost the same performance of PAPR reduction as that of the conventional PTS OFDM scheme.

The authors introduce a new selected mapping (SLM) orthogonal frequency division multiplexing (OFDM) scheme with low computational complexity. The proposed SLM scheme transforms an input symbol sequence into a set of OFDM signals by multiplying the phase sequences to the signal after a certain intermediate stage of inverse fast Fourier transform (IFFT). Then, the OFDM signal with the lowest peak-to-average power ratio (PAPR) is selected for transmission. The new SLM OFDM scheme reduces the computational complexity, while it shows almost the same performance of PAPR reduction as that of the conventional SLM OFDM scheme.

For a prime p and positive integers M and n such that M|p/sup n/-1, Sidel'nikov introduced M-ary sequences (called Sidel'nikov sequences) of period p/sup n/-1, the out-of-phase autocorrelation magnitude of which is upper bounded by 4. In this correspondence, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel'nikov sequences. The frequency of each autocorrelation value of an M-ary Sidel'nikov sequence is expressed in terms of the cyclotomic numbers of order M. It is also pointed out that the total number of distinct autocorrelation values is dependent not only on M but also on the period of the sequence, but always less than or equal to (M/2)+1.

In this paper, given a composite integer n, we propose a method of constructing quaternary low correlation zone (LCZ) sequences of period 2/sup n/-1 from binary sequences of the same length with ideal autocorrelation. These new sequences are optimal with respect to the bound by Tang, Fan, and Matsufuji. The correlation distributions of these new quaternary LCZ sequences constructed from m-sequences and Gordon-Mills-Welch (GMW) sequences are derived.

For a prime power q, we show that a cyclic relative difference set with parameters (q/sup n/-1/q-1,q-1,q/sup n-1/,q/sup n-2/) can be constructed from a d-homogeneous function from F/sub q//sup n//spl bsol/{0} onto F/sub q/ with difference-balanced property, where F/sub q//sup n/ is the finite field with q/sup n/ elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (p/sup n/-1/p/sup l/-1,p/sup l/-1,p/sup n-l/,p/sup n-2l/) from p-ary sequences of period p/sup n/-1 with ideal autocorrelation property introduced by Helleseth and Gong. Using a lifting idea, other new cyclic relative difference sets can be constructed from the Helleseth-Gong (HG) sequences. Also, the 3-ranks and the trace representation of the characteristic sequences of cyclic relative difference sets from a specific class of ternary HG sequences and ternary Lin sequences are derived.

In this paper, using p-ary bent functions defined on vector space over the intermediate finite field, we generalized the construction method of the families of p-ary bent sequences with balanced and optimal correlation properties introduced by Kumar and Moreno for an odd prime p, called generalized p-ary bent sequences.

For an odd prime p and integers n, m, and k such that n=(2m+1)k, a new family of p-ary sequences of period p/sup n/-1 with optimal correlation property is constructed using the p-ary Helleseth-Gong sequences with ideal autocorrelation, where the size of the sequence family is p/sup n/. That is, the maximum nontrivial correlation value R/sub max/ of all pairs of distinct sequences in the family does not exceed p/sup n/2/+1, which means the family has optimal correlation in terms of Welch's lower bound. The symbol distribution of the sequences in the family is enumerated. It is also shown that the linear span of the sequences in the family is (m+2)n except for the m-sequence in the family.

In this article, the linear complexity over F/sub p/ of Lempel-Cohn-Eastman (1977) sequences of period p/sup m/-1 for an odd prime p is determined. For p=3,5, and 7, the exact closed-form expressions for the linear complexity over F/sub p/ of LCE sequences of period p/sup m/-1 are derived. Further, the trace representations for LCE sequences of period p/sup m/-1 for p=3 and 5 are found by computing the values of all Fourier coefficients in F/sub p/ for the sequences.

For a positive integer, n, new families, S and U, of binary sequences of period 2/sup n/-1 with low correlations are proposed, where for some positive integer, e, S is defined for odd n/e and U for even n/e. The family S has four-valued correlations and is a generalization of the family of Gold-like sequences introduced by S. Boztas and P.V. Kumar (see ibid., vol.40, p.532-7, 1994). The family U, which is also a generalization of the sequence family defined by P. Udaya ("Polyphase and frequency hopping sequences obtained from finite rings", Ph.D. dissertation, Dept. Elec. Eng., Indian Inst. Technol., Kanpur, 1992), has six-valued correlations. The relationship between Gold-like sequences and Gold sequences is the same as the relationship between the family S and the family constructed from the binary sequences partially contributed by R. Gold (see ibid., vol.IT-14, p.154-6, 1968), T. Kasami (see Coordinated Sci. Lab., Univ. of Illinois,Urbana-Champaign, Tech. Rep. R-285, AD 632574, 1966), and Welch. Using a lifting idea (No, J.-S. and Kumar, P.V., ibid., vol.35, p.371-9, 1989) for the families S and U, families of binary sequences with the same correlation distributions and large linear span are also constructed.

We generalize the construction method of the family of binary bent sequences introduced by Olsen, Scholtz, and Welch (1982) to obtain a family of generalized binary bent sequences with optimal correlation and balance property by using the modified trace transform. Then, the conventional binary bent sequence becomes a special case of our construction method. Several families of the generalized binary bent sequences are constructed by using the bent functions on the intermediate field. Using some of the generalized binary bent sequences, new families of binary sequences with optimal correlation and balance property can be constructed by the lifting idea similar to No (1988) sequences, which are referred to as binary bent-lifted sequences.

In this paper, for a prime number p, a construction method to generate p-ary d-form sequences with the ideal autocorrelation property is proposed and using the ternary sequences found by Helleseth, Kumar, and Martinsen (2000), ternary d-form sequences with the ideal autocorrelation property are constructed. By combining the methods for generating p-ary extended sequences (a special case of geometric sequences) and p-ary d-form sequences, a construction method of p-ary unified (extended d-form) sequences which also have the ideal autocorrelation property is proposed. This is a very general class of p-ary sequences including the binary and nonbinary extended sequences and the d-form sequences. From the ternary sequences by Helleseth, Kumar, and Martinsen, ternary unified sequences with the ideal autocorrelation property are also generated.

We present a construction for binary sequences {s(t)} of period N=p/sup m/-1 for an odd prime p based on the polynomial (z+1)/sup d/+az/sup d/+b, and discuss them in some cases of parameters p, m, d, a, and b. We show that new sequences from our construction are balanced or almost balanced and have optimal three-level autocorrelation for the case when the polynomial (z+1)/sup d/+z/sup d/+a can be transformed into the form z/sup 2/-c. We also derive the distribution of autocorrelation values they take on. The sequences satisfy constant-on-the-coset property, and we show that there are more than one characteristic phases with constant-on-the-coset property. Some other interesting properties of those sequences are presented. For the cases when the polynomial (z+1)/sup d/+z/sup d/+a cannot be transformed into the form z/sup 2/-c, we performed extensive computer search, and results are summarized. Based on these results, some open problems are formulated.

In this correspondence, the linear span of extended sequences of period 2/sup em/-1 which are constructed from the ideally correlated sequences of period of 2/sup m/-1 is derived. It is also shown that the linear span of cascaded Gordon-Mills-Welch (GMW) sequences can be derived in the same context. As examples, the linear span of extended Legendre sequences and cascaded GMW sequences with triple trace are computed.

We present a construction for binary pseudorandom sequences of period 2/sup m/-1 with ideal autocorrelation property using the polynomial z/sup d/+(z+1)/sup d/. We show that the sequence obtained from the polynomial becomes an m-sequence for certain values of d. We also find a few values of d which yield new binary sequences with ideal autocorrelation property when m is 3k/spl plusmn/1, where k is a positive integer. These new sequences are represented using a trace function and the results are tabulated.

In this correspondence, we present five new classes of binary sequences of period 2/sup n/-1 with ideal autocorrelation. These sequences, which correspond to new cyclic Hadamard difference sets, were found by extensive computer search. Conjectures on the general construction of these sequences are formulated.

We present a construction, in a closed form, for an optimal family of 2/sup m/ binary sequences of period 2/sup 2m/-1 with respect to Welch's (1974) bound, whenever there exists a balanced binary sequence of period 2/sup m/-1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No (1988) sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method.

GMW sequences and families of No sequences are generalized. Generalized GMW sequences have ideal autocorrelation and balance properties and generalized No sequences also have optimal correlation properties in terms of Welch's lower bound. The linear spans of the generalized GMW and No sequences appear to be large although there is not at present a closed-form expression for the linear span. A count of the numbers of cyclically distinct generalized GMW sequences and generalized No sequences that can be constructed is provided. Generalized GMW sequences have also been found in the literature by Klapper et al. (1993) under the name "cascaded GMW sequences".

In this correspondence, it is shown that Legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2/sup n/ elements, whenever p=2/sup n/-1 is prime for some n/spl ges/3.

A collection of families of binary