The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.

In this study, the authors present new four‐dimensional signal constellations constructed based on combinations of binary frequency shift keying and M ‐ary phase‐shift keying. Such a system contains two sub‐constellations and phase shift keying modulation is carried out based on two different frequencies. They obtained an analytical expression for calculating of squared Euclidean distance for different signal constellations. The new method of system construction and optimisation is presented with the help of this formula. Having used this method for various signal constellations and modulation indexes, they constructed optimised systems. In tables, they present the parameters for signal constellation sizes 4, 5, …,16 and modulation indexes 0.1, 0.2, …,1. Such signal constellations can be used in MIMO systems and, in particular, in generalised multi‐stream spatial modulation systems.

Error correction coding based on soft‐input decoding can significantly improve the reliability of non‐volatile flash memories. This work proposes a soft‐input decoder for generalised concatenated (GC) codes. GC codes are well suited for error correction in flash memories for high reliability data storage. The authors propose GC codes constructed from inner extended binary Bose–Chaudhuri–Hocquenghem (BCH) codes and outer Reed–Solomon (RS) codes. The extended BCH codes enable an efficient hard‐input decoding. Furthermore, a low‐complexity soft‐input decoding method is proposed. This bit‐flipping decoder uses a fixed number of test patterns and an algebraic decoder for soft‐decoding. An acceptance criterion for the final candidate codeword is proposed. Combined with error and erasure decoding of the outer RS codes, this acceptance criterion can improve the decoding performance and reduce the decoding complexity. The presented simulation results show that the proposed bit‐flipping decoder in combination with outer error and erasure decoding can outperform maximum‐likelihood decoding of the inner codes.

Generalized concatenated (GC) codes with soft-input decoding were recently proposed for error correction in flash memories. This work proposes a soft-input decoder for GC codes that is based on a low-complexity bit-flipping procedure. This bit-flipping decoder uses a fixed number of test patterns and an algebraic decoder for soft-input decoding. An acceptance criterion for the final candidate codeword is proposed. Combined with error and erasure decoding of the outer Reed-Solomon codes, this bit-flipping decoder can improve the decoding performance and reduce the decoding complexity compared to the previously proposed sequential decoding. The bit-flipping decoder achieves a decoding performance similar to a maximum likelihood decoder for the inner codes.

This work proposes a construction for low-density parity-check (LDPC) codes over finite Gaussian integer fields. Furthermore, a new channel model for codes over Gaussian integers is introduced and its channel capacity is derived. This channel can be considered as a first order approximation of the additive white Gaussian noise channel with hard decision detection where only errors to nearest neighbors in the signal constellation are considered. For this channel, the proposed LDPC codes can be decoded with a simple non-probabilistic iterative decoding algorithm similar to Gallager's decoding algorithm A.

This paper proposes a soft input decoding algorithm and a decoder architecture for generalized concatenated (GC) codes. The GC codes are constructed from inner nested binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer Reed-Solomon codes. In order to enable soft input decoding for the inner BCH block codes, a sequential stack decoding algorithm is used. Ordinary stack decoding of binary block codes requires the complete trellis of the code. In this paper, a representation of the block codes based on the trellises of supercodes is proposed in order to reduce the memory requirements for the representation of the BCH codes. This enables an efficient hardware implementation. The results for the decoding performance of the overall GC code are presented. Furthermore, a hardware architecture of the GC decoder is proposed. The proposed decoder is well suited for applications that require very low residual error rates.

Codes over quotient rings of Lipschitz integers have recently attracted some attention. This work investigates the performance of Lipschitz integer constellations for transmission over the AWGN channel by means of the constellation figure of merit. A construction of sets of Lipschitz integers that leads to a better constellation figure of merit compared to ordinary Lipschitz integer constellations is presented. In particular, it is demonstrated that the concept of set partitioning can be applied to quotient rings of Lipschitz integers where the number of elements is not a prime number. It is shown that it is always possible to partition such quotient rings into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is strictly larger than in the original set. The resulting signal constellations have a better performance for transmission over an additive white Gaussian noise channel compared to Gaussian integer constellations and to ordinary Lipschitz integer constellations. In addition, we present multilevel code constructions for the new signal constellations.

Codes over Gaussian integers have been proposed for coding over two‐dimensional (2D) signal spaces, for example, using quadrature amplitude modulation. Here, it is demonstrated that the concept of set partitioning can be applied to Gaussian integer constellations that are isomorphic to 2D modules over rings of integers modulo p . This enables multilevel code constructions over Gaussian integers. The authors derive upper bounds on the achievable minimum distance in the subsets and present a construction for the set partitioning. This construction achieves optimal or close to optimal minimum distances. Furthermore, it is demonstrated that this set partitioning can be applied to an interleaving technique for correcting 2D cyclic clusters of errors. The authors propose a novel combination of generalised concatenated codes with 2D interleaving to correct 2D error clusters and independent errors.

This work presents block codes over Gaussian integers. We introduce Gaussian integer rings which extend the number of possible signal constellations over Gaussian integer fields. Many well-known code constructions can be used for codes over Gaussian integer rings, e.g., the Plotkin construction or product codes. These codes enable low complexity decoding in the complex domain. Furthermore, we demonstrate that the concept of set partitioning can be applied to Gaussian integers. This enables multilevel code constructions. In addition to the code constructions, we present a low complexity soft-input decoding algorithm for one Mannheim error correcting codes. The presented decoding method is based on list decoding, where the list of candidate codewords is obtained by decomposing the syndrome into two sub-syndromes. Considering all decompositions of the syndrome we construct lists of all possible errors of Mannheim weight two. In the last decoding step the squared Euclidean distance is used to select the best codeword from the list. Simulation results for the additive white Gaussian noise channel demonstrate that the proposed decoding method achieves a significant coding gain compared with hard-input decoding.

This paper addresses space time convolutional code design using continuous phase modulation (CPM). The possibility of constructing full diversity space time codes is investigated. A linear modulation approximation to CPM is done. Using the Gram-Schmidt orthogonalization transform the CPM signal is generated as a vector with finite energy in a different Euclidean space. A serially concatenated CPM construction is considered in searching channel codes which are able to exploit maximum diversity. Design criteria based on the encoding scheme are derived for an arbitrary number of transmit antennas. The investigations are done for a quasi-static Rayleigh fading channel.

We introduce hierarchical woven coded continuous phase frequency-shift keying (hierarchical WCCPFSK) as the serial concatenation of different outer convolutional codes and inner CPFSK. We compare it to WCCPFSK with identical outer convolutional codes. With the proposed code combinations, hierarchical WCCPFSK achieves superior decoding capability. Simulations show that it performs better at medium SNRs.

In this correspondence, we investigate a decision feedback strategy for convolutional codes which is based on a sliding window decoding procedure and a threshold test as decision rule. For this purpose, we introduce the burst distance spectrum of a convolutional code and derive asymptotic bounds for the ensemble of periodically time-varying convolutional codes. These results are helpful for the asymptotic analysis of the decision feedback scheme. We show that unit memory codes are particularly suited for such a transmission scheme. For these codes, the decoding procedure is reduced to the decoding of block codes with lengths in the order of the overall constraint length of the convolutional code. This leads to a significantly smaller decoding complexity compared with other known decoding and decision rules. Whereas the achievable asymptotic performance is close to the best known bounds. For low rates, our results even improve these bounds.

We investigate a decision feedback strategy for convolutional codes which is based on a sliding window decoding procedure and a threshold test as decision rule. For this purpose, we introduce the burst distance spectrum of a convolutional code and derive asymptotic bounds for the ensemble of periodically time-varying convolutional codes. These results are helpful for the asymptotic analysis of the decision feedback scheme. Unit memory codes are particularly suited for such a transmission scheme. For these codes, the decoding procedure is reduced to the decoding of block codes with lengths in the order of the overall constraint length of the convolutional code. This leads to a significantly smaller decoding complexity compared with other known decision rules. Whereas the achievable asymptotic performance is close to the best known bounds. For low rates, our results even improve these bounds.

This paper describes the hierarchical woven coded CPFSK (hWCCPFSK), is a serial concatenation of different outer convolutional codes and inner continuous phase frequency shift keying (CPFSK). Classical woven coded CPFSK (cWCCPFSK) has identical outer convolutional codes. We show that with a proper choice of the outer codes, hWCCPFSK has better decoding behaviour but cWCCPFSK has larger free distance. Hence, hWCCPFSK performes better in the waterfall region of the bit error rate curves, while cWCCPFSK is better in the error floor region.

We present a new concatenated code construction. The resulting codes can be viewed as intermediate between parallel and serially concatenated convolutional codes. Proper partitioning of the outer code sequence provides a new degree of freedom for code design. Various methods are considered to analyze code properties.

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties.

In this correspondence, the error exponents and decoding complexity of binary woven convolutional codes with outer and inner warp are studied. It is shown that for both constructions an error probability that is exponentially decreasing with the memory of the woven convolutional codes can be achieved with a nonexponentially increasing decoding complexity. Furthermore, the error exponent for woven convolutional codes with inner warp is larger than the one for woven convolutional codes with outer warp.

In this correspondence the error exponent and the decoding complexity of binary woven convolutional codes with outer warp and with binary convolutional codes as outer and inner codes are studied. It is shown that an error probability that is exponentially decreasing with the product of the outer and inner code memories can be achieved with a nonexponentially increasing decoding complexity.

A novel construction for encoded tamed frequency modulation (TFM) is introduced which is based on the principles of generalized concatenation. The inner TFM is partitioned into nested subsystems which increases the free Euclidean distances. In order to obtain a large distance among the nested TFM subsystems, the scrambler matrices have to be computed which transfer the original TFM into the equivalent TFM with better partitioning properties. Then outer convolutional codes with different error-correcting capabilities are used to protect the partitioning. The new concatenated and generalized concatenated constructions were simulated in an additive white Gaussian noise channel. A multistep decoding algorithm based on soft-output demodulation was used. We present various simulation results which show a significant coding gain in comparison with the best known trellis codes having the same trellis state complexity.