In this letter, we propose a communication efficient federated learning algorithm, coined random FLARE (R-FLARE), using a novel error compensation method within a framework of random sparsification. In the R-FLARE, all devices sparsify the local gradients using a common set of randomly selected indices to improve communication efficiency with over-the-air computation. To upload local gradients, only the selected gradient elements are compensated by the local errors accumulated due to sparsification, which prevents redundant error compensation. We conduct a theoretical analysis on the convergence of R-FLARE using the$l_{2}$norm-based error compensation, which shows that it achieves the same convergence rate as the state-of-the-art algorithms. Numerical results show that the R-FLARE using$l_{1}$- and$l_{2}$-norm based error compensations outperform conventional algorithms in test accuracy and training speed.

Reconfigurable intelligent surface (RIS) is a key enabler for 6G wireless networks, reconfiguring wireless propagation to enhance the quality of signal transmission. In RIS-aided communications, channel estimation plays a crucial role because the performance improvement from RIS relies on the accuracy of channel state information. In this paper, we configure the phase shifts of RIS passive reflective elements, in an effort to improve the performance of cascaded channel estimation between a base station and users through the RIS in multiuser mmWave communications. We first apply the Golay complementary sequences to configure the bipolar phase shifts of RIS in training phase. Then, we study a theoretical recovery guarantee for compressed sensing based channel estimation with the Golay-based bipolar phase shifts of RIS. Simulation results demonstrate that the Golay-based phase shift configuration exhibits excellent performance for channel estimation, outperforming random and optimized phase shifts. The algebraic configuration of phase shifts contributes to improving the performance of channel estimation efficiently through a large-scale RIS with bi-phase resolution in RIS-aided multiuser mmWave communications.

Grant-free access is a key enabler for connecting wireless devices with low latency and low signaling overhead in massive machine-type communications (mMTC). For massive grant-free access, user-specific signatures are uniquely assigned to mMTC devices. In this paper, we first derive a sufficient condition for the successful identification of active devices through maximum likelihood (ML) estimation in massive grant-free access. The condition is represented by the coherence of a signature sequence matrix containing the signatures of all devices. Then, we present a design framework of non-orthogonal signature sequences in a deterministic fashion. The design principle relies on unimodular masking sequences with low correlation, which are applied as masking sequences to the columns of the discrete Fourier transform (DFT) matrix. For example constructions, we use four polyphase masking sequences represented by characters over finite fields. Leveraging algebraic techniques, we show that the signature sequence matrix of proposed non-orthogonal sequences has theoretically bounded low coherence. Simulation results demonstrate that the deterministic non-orthogonal signatures achieve the excellent performance of joint activity and data detection by ML- and approximate message passing (AMP)-based algorithms for massive grant-free access in mMTC.

In edge intelligence, deep learning (DL) models are deployed at an edge device and an edge server for data processing with low latency in the Internet of Things (IoT). In this paper, we propose a new end-to-end learning-based wireless image recognition scheme using the PyramidNet in edge intelligence. We split the PyramidNet carefully into two parts for an IoT device and the edge server, which is to pursue low on-device computation. Also, we apply a squeeze-and-excitation block to the PyramidNet for the improvement of image recognition. In addition, we embed compression encoder and decoder at the splitting point, which reduces communication overhead by compressing the intermediate feature map. Simulation results demonstrate that the proposed scheme is superior to other DL-based schemes in image recognition, while presenting less on-device computation and fewer parameters with low communication overhead.

Non-orthogonal multiple access (NOMA) is an emerging technology for massive connectivity in machine-type communications (MTC). In code-domain NOMA, non-orthogonal spreading sequences are uniquely assigned to all devices, where active ones attempt a grant-free access to a system. In this paper, we study a set of user-specific, non-orthogonal, binary spreading sequences for uplink grant-free NOMA. Based on binary Golay complementary sequences, each spreading sequence provides the peak-to-average power ratio (PAPR) of at most 3 dB for multicarrier transmission. Exploiting the theoretical connection to Reed-Muller codes, we conduct a probabilistic analysis to search for a permutation set for Golay sequences, which presents theoretically bounded low coherence for the spreading matrix. Simulation results confirm that the maximum PAPR of transmitted multicarrier signals via the spreading sequences is significantly lower than those for random bipolar, Gaussian, and pseudo-random sequences. Thanks to the low coherence, the performance of compressed sensing (CS) based joint channel estimation and multiuser detection using the spreading sequences turns out to be superior or similar to those for random bipolar, Gaussian, pseudo-random, and Zadoff-Chu sequences. The binary Golay spreading sequences have only two phases regardless of the sequence length, which can make them suitable for low cost MTC devices.

In this paper, we study the security of a compressed sensing (CS) based cryptosystem called a sparse one-time sensing (S-OTS) cryptosystem, which encrypts a plaintext with a sparse measurement matrix. To construct the secret matrix and renew it at each encryption, a bipolar keystream and a random permutation pattern are employed as cryptographic primitives, which can be obtained by a keystream generator of stream ciphers. With a small number of nonzero elements in the measurement matrix, the S-OTS cryptosystem achieves efficient CS encryption in terms of memory and computational cost. In security analysis, we show that the S-OTS cryptosystem can be indistinguishable as long as each plaintext has constant energy, which formalizes computational security against ciphertext only attacks (COA). In addition, we consider a chosen plaintext attack (CPA) against the S-OTS cryptosystem, which consists of two sequential stages, keystream and key recovery attacks. Against keystream recovery under CPA, we demonstrate that the S-OTS cryptosystem can be secure with overwhelmingly high probability, as an adversary needs to distinguish a prohibitively large number of candidate keystreams. Finally, we conduct an information-theoretic analysis to show that the S-OTS cryptosystem can be resistant against key recovery under CPA by guaranteeing that the probability of success is extremely low. In conclusion, the S-OTS cryptosystem can be computationally secure against COA and the two-stage CPA, while providing efficiency in CS encryption.

In this paper, we study the security of a compressed sensing (CS) based cryptosystem that encrypts a plaintext with a sparse measurement matrix. The secret matrix is constructed by a bipolar keystream and a random permutation, and renewed at each encryption. The CS-based cryptosystem performs efficient encryption with a small number of nonzero entries in the matrix, and guarantees reliable decryption for a legitimate recipient. Through a quantitative analysis, we demonstrate that the CS-based cryptosystem achieves the security against a chosen plaintext attack (CPA) with overwhelmingly high probability, by showing that an adversary needs to distinguish a prohibitively large number of candidate keystreams.

In heterogeneous ultra-dense networks with millimeter wave macro cells and small cells, base stations (BSs) and mobile user equipments (UEs) perform beamforming operations to establish highly directional links. In spite of the spatial diversity achieved through directional links, as a number of BSs are densely deployed, inter-cell interference caused by concurrent directional transmissions of adjacent BSs becomes severe, resulting in downlink performance degradation in the network. However, it is very difficult to manage inter-cell interference because of the nature of the time-varying wireless fading environment, the dynamic changes in beam propagation directivity, and unpredictable UEs' locations. In this paper, we propose an online learning-based transmission coordination algorithm based on the framework of multi-armed bandits to learn the unknown characteristics of inter-BS interference and exploit learned data to derive an optimal policy for maximizing the number of successful downlink transmissions. Through the numerical simulations, we verify the effectiveness of the proposed online learning-based inter-BS interference management scheme.

In this letter, we study the security of a cryptosystem over wireless channels that employs the asymptotically Gaussian compressed encryption. We investigate the indistinguishability and the energy sensitivity of the cryptosystem, where the total variation (TV) distance is examined as a statistical measure for the indistinguishability. To characterize the TV distance, we compute the Hellinger distance between probability distributions of ciphertexts, each of which can be modeled as a circularly symmetric complex Gaussian random vector with a constraint on plaintexts. Using the distance metrics, we show that the cryptosystem can be a promising option for secure wireless communications by guaranteeing the indistinguishability against an eavesdropper, as long as each plain text has constant energy.

The principle of compressed sensing (CS) can be applied in a cryptosystem by providing the notion of security. The Gaussian one-time sensing (G-OTS) CS-based cryptosystem employing a random Gaussian matrix and renewing the elements at each encryption is known to be perfectly secure, as long as each plaintext has constant energy. A random Bernoulli matrix can replace the Gaussian one for encrypting each plaintext efficiently in the Bernoulli one-time sensing (B-OTS) cryptosystem. In this paper, we analyze the security of G-OTS and B-OTS cryptosystems, respectively, where each cryptosystem may have unequal plaintext energy. By means of probability metrics, we study the indistinguishability of each CS-based cryptosystem, which formalizes the notion of computational security. Moreover, we investigate how much the indistinguishability is sensitive to energy variation of plaintexts in each cryptosystem. For the B-OTS cryptosystem, we analyze the indistinguishability and the energy sensitivity in a non-asymptotic manner for a finite plaintext length. In conclusion, this paper confirms that G-OTS and B-OTS cryptosystems can be strictly and asymptotically indistinguishable, respectively, as long as each plaintext has constant energy, but the indistinguishability is highly sensitive to energy variation of plaintexts.

In this paper, we study random access for machinetype communications (MTC) based on a multiple access scheme that exploits different channel state information (CSI) from active devices under frequency-selective fading, which is called channel division multiple access (ChDMA). In ChDMA, in order to use a low-complexity compressive sensing (CS) algorithm for multiuser detection (MUD) with a high transmission rate, we employ index modulation. Since the CSI is used as a signature for MUD, the CSI estimation becomes crucial, but suffers from the pilot collision. In order to lower the probability of pilot collisions, multiple transmissions of randomly selected pilots are considered.

In machine-type communications (MTC), a number of devices are present in a cell, but only a few of them attempt to access by sending preambles or pilots. For the sparse activity of devices, compressive sensing (CS) can be applied for joint detection of active devices and estimation of channel profiles. In this paper, a theoretical connection between pilots and the coherence of a CS measurement matrix is revealed for CS-based detection and estimation. Then, the Zadoff-Chu (ZC) and the power residue (PR) sequences are studied as potential candidates for pilot signals in CS-based random access. It is also shown that the CS measurement matrices from the ZC- and the PR-based pilots have theoretically bounded low coherence, which presents a theoretical guarantee of reliable recovery performance. Simulation results demonstrate that the proposed deterministic pilots outperform random binary and complex-valued pilots for CS-based random access in MTC.

The principle of compressed sensing (CS) can be applied to a cryptosystem in which the sensing matrix is employed for the secret key. In this letter, we study the security of a CS-based cryptosystem that encrypts a plaintext with a secret circulant matrix and transmits the ciphertext over a wireless channel. The relative entropy is considered as a security measure for the indistinguishability of the CS-based cryptosystem. By developing an upper bound on the entropy, the security analysis reveals that the presence of wireless channels and additive noise contributes to reducing the relative entropy of the cryptosystem. Consequently, the CS-based cryptosystem with circulant matrices can guarantee wireless security in terms of the indistinguishability, as long as the channel gains and the plaintext-to-noise power ratio of an adversary are kept to be low for a long keystream and a short ciphertext.

In this paper, we study random access for machine-type communications based on a multiple access scheme that exploits different channel state information (CSI) from active devices under frequency-selective fading, which is called channel division multiple access (ChDMA). In ChDMA, in order to use a low-complexity compressive sensing (CS) algorithm for multiuser detection (MUD) with a high transmission rate, we propose to use index modulation. Since the CSI is used as a signature for MUD, the CSI estimation becomes crucial. For the CSI estimation, pilot transmission can be considered where each active device randomly chooses a pilot from a set of pre-determined pilots. Since the CSI estimation suffers from pilot collision in this case, we propose multiple pilot transmissions and show that a low probability of collision can be achieved. A CS-based approach is derived not only to estimate the CSI, but also to detect randomly selected multiple pilot signals by exploiting the sparsity of active devices over frequency-selective fading channels.

In this paper, we study a support set reconstruction problem for multiple measurement vectors (MMV) with different sensing matrices, where the signals of interest are assumed to be jointly sparse and each signal is sampled by its own sensing matrix in the presence of noise. Using mathematical tools, we develop upper and lower bounds of the failure probability of the support set reconstruction in terms of the sparsity, the ambient dimension, the minimum signal-to-noise ratio, the number of measurement vectors, and the number of measurements. These bounds can be used to provide guidelines for determining the system parameters for various compressed sensing applications with noisy MMV with different sensing matrices. Based on the bounds, we develop necessary and sufficient conditions for reliable support set reconstruction. We interpret these conditions to provide theoretical explanations regarding the benefits of taking more measurement vectors. We then compare our sufficient condition with the existing results for noisy MMV with the same sensing matrix. As a result, we show that noisy MMV with different sensing matrices may require fewer measurements for reliable support set reconstruction, under a sublinear sparsity regime in a low noise-level scenario.

Digital fingerprinting has been proposed to restrict illegal distribution of digital media, where every piece of media has a unique fingerprint as an identifying feature that can be traceable. However, fingerprint systems are vulnerable when multiple users collude to create a forged copy by combining their ones. The collusion is modeled as a linear averaging attack, where multiple weighted copies are averaged and the Gaussian noise is then added to the averaged copy. In this paper, a new fingerprint design robust to collusion is proposed, which is to accommodate a lot more users than other existing fingerprint designs. A base matrix is constructed by cyclic shifts of binary sequences in an optical orthogonal code and then extended by a Hadamard matrix. Finally, each column of the resulting matrix will be used as a fingerprint. The focused detection is used to determine whether a user is innocent or guilty in a collusion of linear averaging attack. Simulation results show that the performance of our new fingerprint design is comparable to that of orthogonal and simplex fingerprints.

In many applications of compressed sensing, the data acquisition involves convolution by a filter followed by subsampling. In this letter, we propose to construct a filter with real-valued coefficients by taking the discrete Fourier transform of a decimated binary Sidelnikov sequence. With a random subsampler, we prove that stable recovery can be guaranteed if a signal is sparse in the canonical or the FFT basis. Besides, simulation results also show that if a deterministic subsampler is used, the proposed system can offer similar reconstruction performance as that of a random Gaussian operator for a wide range of signal length.

In this letter, a new class of real-valued matrices is presented for deterministic compressed sensing. A base matrix is constructed by cyclic shifts of binary sequences in an optical orthogonal code (OOC). Then, a Hadamard matrix is used for its extension, which ultimately produces a real-valued matrix that takes the entries of 0, -1 and +1 before normalization. The new sensing matrix forms a tight frame with small coherence, which theoretically guarantees the average recovery performance of sparse signals with uniformly distributed supports. Several example sensing matrices are presented by employing a special type of OOCs obtained from modular Golomb rulers.

In this paper, the sequence families of which maximum correlation is determined by the Weil bound of exponential sums are revisited. Using the same approach, two new constructions with large family sizes and low maximum correlation are given. The first construction is an analog of one recent result derived from the interleaved structure of Sidel'nikov sequences. For a primepand an integerM|(p-1), the newM-ary sequence families of periodpare obtained from irreducible quadratic polynomials and known power residue-based sequence families. The new sequence families increase family sizes of the known power residue-based sequence families, but keep the maximum correlation unchanged. In the second construction, the sequences derived from the Weil representation are generalized, where each new sequence is the elementwise product of a modulated Sidel'nikov sequence and a modulated trace sequence. For positive integersdpandM|(pn-1), the new family consists of (M-1)pndsequences with periodpn-1, alphabet sizeMp, and the maximum correlation bounded by (d+1)√{pn}+3.

In this paper, an (N, K) codebook is constructed from a K × N partial matrix with K <; N, where each code vector is equivalent to a column of the matrix. To obtain the K × N matrix, K rows are selected from a J × N matrix Φ, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of Φ-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its Φ-transform is as small as possible. From the discovery, new classes of near-optimal codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary Sidelnikov sequences.

A codebook, or equivalently a frame, has many applications in communications, signal processing, and quantum computing. In the applications, it is required that the maximum magnitude of inner products between a pair of distinct code vectors should meet the Welch bound equality strictly or asymptotically. In this paper, a new (N, K) codebook is constructed from a K×N partial Fourier matrix with N = K2-1 and K=2kfor a positive integer k, where each code vector is equivalent to a column of the matrix. To obtain the K×N partial Fourier matrix, K rows are selected from the N ×N inverse discrete Fourier transform matrix, where a set of the selected row indices is determined from the structure of binary m-sequences of period N. Through a theoretical study, it is found that the magnitude of inner products between distinct code vectors is two-valued, and its maximum nearly achieves the Welch bound equality. The new near-optimal codebook, equivalent to a nearly equiangular tight frame, can find its potential applications in deterministic compressed sensing.

An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. For its applications, it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible, meeting the Welch bound equality strictly or asymptotically. In this paper, an (N, K) codebook is constructed from a K × N partial matrix with K <; N, where each code vector is equivalent to a column of the matrix. To obtain the K × N matrix, K rows are selected from a J × N matrix Φ, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of Φ-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its Φ-transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this paper, a K × N measurement matrix for compressed sensing is deterministically constructed via additive character sequences. The Weil bound is then used to show that the matrix has asymptotically optimal coherence for N = K2, and that it is a tight frame. A sparse recovery guarantee for the incoherent tight frame is also discussed. Numerical results show that the deterministic sensing matrix guarantees empirically reliable recovery performance via an l1-minimization method for noiseless measurements.

New families of near-complementary sequences are presented for peak power control in multicarrier communications. A framework for near-complementary sequences is given by an explicit Boolean expression and an equivalent matrix structure. The framework transforms seed pairs to near-complementary sequences by the aid of Golay complementary sequences. New families of near-complementary sequences of various lengths and PMEPR <; 4 are then presented, where the sequences are constructed by the framework employing the seeds of shortened and extended Golay complementary pairs. The families present in a constructive way a large number of sequences of PMEPR <; 4 for the lengths (<; 100) of 24, 28, 30, 34, 36, 48, 56, 60, 62, 66, 68, 72, and 96 where no Golay pairs have been reported. The sequence families can find potential applications for peak power control requiring codewords or sequences of various lengths as well as low peak-to-mean envelope power ratio (PMEPR).

For prime p and a positive integer m, it is shown that M-ary Sidelnikov sequences of period p2m- 1, if M | pm- 1, can be equivalently generated by the operation of elements in a finite field GF(pm), including a pm-ary m-sequence. The equivalent representation over GF(pm) requires low complexity for implementing the Sidelnikov sequences. Moreover, a (pm- 1)×(pm+1) array structure is introduced for the Sidelnikov sequences. From the array structure, it is found that about a half of the column sequences of length pm- 1 and their constant multiples have the low correlation magnitude bounded by 3√(pm+ 1).

In this paper, generalized constructions of polyphase sequence families from the shift and addition of power residue and Sidelnikov sequences are presented. Initially, ψ(0) = 1 is assumed for multiplicative characters ψ to represent power residue and Sidelnikov sequences in a simple form. The Weil bound on multiplicative character sums is refined for the assumption, where the character sums are equivalent to the correlations of sequences represented by multiplicative characters. Generalized constructions are then presented by the addition of multiple cyclic shifts of power residue and Sidelnikov sequences. The refined Weil bound is employed to provide efficient proofs on the maximum correlation magnitudes of the generalized sequence families.

For primepand a positive integerm, it is shown thatM-ary Sidelnikov sequences of periodp2m-1, ifM|pm-1, can be equivalently generated by the operation of elements in a finite fieldGF(pm), including apm-arym-sequence. From the(pm-1) ×(pm+1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct newM-ary sequence families of periodpm-1. In particular, newM-ary sequence families of periodpm-1 are constructed from the combination of the column sequence families and known Sidelnikov-based sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the newM-ary sequence family of periodpm-1 and the maximum correlation magnitude2√{pm}+6 asymptotically achieves√2times the equality of the Sidelnikov's lower bound whenM=pm-1 for odd primep.

Power residue and Sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes, represented by multiplicative characters. In this paper, sequence families constructed from the shift and addition of the polyphase sequences are revisited. Initially, ψ(0)=1 is assumed for multiplicative characters ψ to represent power residue and Sidelnikov sequences in a simple form. The Weil bound on multiplicative character sums is refined for the assumption, where the character sums are equivalent to the correlations of sequences represented by multiplicative characters. General constructions of polyphase sequence families that produce some of known families as the special cases are then presented. The refined Weil bound enables the efficient proofs on the maximum correlation magnitudes of the sequence families. From the constructions, it is shown that M-ary known sequence families with large size can be partitioned into (M+1) disjoint subsequence families with smaller maximum correlation magnitudes. More generalized constructions are also considered by the addition of multiple cyclic shifts of power residue and Sidelnikov sequences.

New families of near-complementary sequences are presented for peak power control in multicarrier communications. A framework for near-complementary sequences is given by the explicit Boolean expression and the equivalent array structure. The framework transforms the seed pairs to the near-complementary sequences by the aid of Golay complementary sequences. As the examples, new families of near-complementary sequences with the peak-to-mean envelope power ratio (PMEPR) < 4 are presented, where the sequences provide various lengths by employing the seeds of shortened or extended Golay complementary pairs. The sequence families can find the potential applications for peak power control requiring codewords or sequences of various lengths as well as low PMEPR.

Performances of punctured tail-biting convolutional codes are studied using a novel initial state estimation technique. The initial states of a tail-biting trellis are estimated as soft initial metrics by channel observations and code configurations for improving the performance of the circular Viterbi decoding of tail-biting convolutional codes. The initial state estimation is applied for any kind of systematic and a special form of nonsystematic tail-biting convolutional codes. Through simulations, it is observed that the initial state estimation improves the decoding performances of punctured tail-biting convolutional codes in the circular Viterbi decoder with delay constraints.

New binary sequences of period N = 4(2m- 1) for even m ges 4 are found, where the sequences are described by a 4 X (2m- 1) array structure. The new sequences are almost balanced and have four- valued autocorrelation, i.e., {N, 0, plusmn4}, which is optimal with respect to autocorrelation magnitude. The complete autocorrelation distribution and the exact linear complexity of the sequences are mathematically derived. Finally, it is shown that the sequences are implemented by a combination of linear feedback shift registers and a simple logic.

For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/

In this correspondence, the constructions and enumerations of all bent functions represented by a polynomial form of f(x)=/spl Sigma//sub i=1//sup n/2-1/c/sub i/Tr(x/sup 1+2(i)/)+c/sub n/2/Tr/sub 1//sup n/ /sup /2/(x/sup 1+2(n/2)/), c/sub i//spl isin//sub 2/ F/sub 2/ are presented for special cases of n. Using an iterative approach, the construction of bent functions of n variables with degree n/2 is also provided using the constructed quadratic bent functions.

Multiple access interference (MAI) in CDMA system is related to the correlation of the transmitted signals from multiple users. In this work, we investigate the correlations of binary signal sets over fading channels, particularly the Gold and Kasami signal sets. A lower bound on the maximum correlation with fading is established. Moreover, an asymptotic bound depending on the length of the sequence and the first and second order statistics of the fading distribution is derived for independent fading channel, while an approximation to the maximum correlation is obtained for correlative fading channel. To analyze the fading effect on cross correlation, experimental results show that the distribution of cross correlation in slow fading environment is similar to the one without fading.

Iterative decoding of product codes that uses a soft-in/soft-out decoder based on the Chase algorithm was introduced by R. Pyndiah et al. (see Proc. IEEE GLOBECOM, vol.1/3, p.339-43, 1994). This decoding scheme is quite similar to turbo codes, but makes possible the implementation of the decoder with low complexity while providing acceptable performance. In this paper for the product codes composed of extended Hamming codes, we propose a reduction scheme of test patterns in the Chase algorithm, which can reduce the decoding complexity without performance degradation. Also, we propose a simple, but efficient assignment scheme of artificial extrinsic information to the symbols for which we cannot get reliabilities by the decoding procedure. The simulation results for the proposed schemes are shown and compared with the asymptotic performances that can be calculated by the weight distribution of product codes and the union bound on the BER performances.