Nonbinary Block Codes Research Articles

Constellation Design for Media-Based Modulation Using Block Codes and Squaring Construction

Efficient constellation design is important for improving performance in communication systems. The problem of multidimensional constellation design has been studied extensively in the literature in the context of multidimensional coded modulation and space-time coded MIMO systems. Such constellations are formally called as lattice codes, where a finite set of points from a certain high dimensional lattice is chosen based on some criteria. In this paper, we consider the problem of constellation/signal set design for media-based modulation (MBM), a recent MIMO channel modulation scheme with promising theoretical and practical benefits. Constellation design for MBM is fundamentally different from those for multidimensional coded modulation and conventional MIMO systems mainly because of the inherent sparse structure of the MBM signal vectors. Specifically, we need a structured sparse lattice code with good distance properties. In this work, we show that using an (N,K) non-binary block code in conjunction with the lattice based multilevel squaring construction, it is possible to systematically construct a signal set for MBM with certain guaranteed minimum distance. The MBM signal set obtained using the proposed construction is shown to achieve significantly improved bit error performance compared to conventional MBM signal set. In particular, the proposed signal set is found to achieve higher diversity slopes in the low-to-moderate SNR regime.

Sequential Decoding on Syndrome Trellises for Nonparametric Detection

Abstract The following paper adapts the classical Zigangirov-Jelinek algorithm to the decoding of nonbinary block codes under severe mixed jamming. To ensure reliable communications in this scenario we combine reception techniques based on distribution free statistical tests with sequential decoding on syndrome trellises. It will be shown that the proposed approach can ensure relatively high transmission rate with reasonable complexity.

Searching for Binary and Nonbinary Block and Convolutional LDPC Codes

A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF $(2^{m})$ , we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable $D$ , we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate $R=1/2$ LDPC code of length 64 800 optimized for effective data rate $R=3/4$ gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards.

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

Reconstruction of a Non-binary Block Code from an Intercepted Sequence with Application to Reed-Solomon Codes

In this paper, a method is proposed for reconstruction of the parameters of a non-binary block encoder using an intercepted sequence of noisy coded data. The proposed method is a generalization of the Barbier's method for the reconstruction of binary block codes to the more problematic case of non-binary codes. It has been shown mathematically that considering some revisions in definitions, such a generalization is possible. The proposed method is able to estimate the code parameters such as the code length, the code dimension, number of bits per symbol, and the dual-code subspace, and also to synchronize the sequence. Since the Reed-Solomon code is the most important type of non-binary block codes, an additional method is proposed to reconstruct the generator polynomial in the case of Reed-Solomon codes. The proposed method is evaluated via computer simulations which verify its strength and effectiveness.

On Symbol Versus Bit Interleaving for Block-Coded Binary Markov Channels

We examine bit- and symbol-interleaving strategies for linear nonbinary block codes (under bounded-distance decoding) over the family of binary additive noise finite-state Markov channel (FSMC) models with memory. We derive a simple analytical sufficient condition under which perfect (i.e., with infinite interleaving depth) symbol interleaving outperforms perfect bit interleaving in terms of the probability of codeword error (PCE). It is shown that the well-known Gilbert-Elliott channel (GEC) with positive noise-correlation coefficient and the recently introduced Markovian queue-based channel (QBC) of memory M satisfy this condition. This result has widely been illustrated numerically (without proof) in the literature, particularly for the GEC. We also provide examples of binary FSMC models for which the reverse result holds, i.e., perfect bit interleaving outperforming perfect symbol interleaving. Finally, a numerical PCE study of imperfect symbol-interleaved nonbinary codes over the QBC indicates that there is a linear relationship between the optimal interleaving depth and a function of a single parameter of the QBC.

Performance Bounds for Nonbinary Linear Block Codes Over Memoryless Symmetric Channels

The performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum-likelihood (ML) decoding. The transmission of these codes is assumed to take place over a memoryless and symmetric channel. The new bounds, which are based on the Gallager bounds and their variations, are applied to the Gallager ensembles of nonbinary and regular low-density parity-check (LDPC) codes. These upper bounds are also compared with sphere-packing lower bounds. This study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques.

Selection of Test Patterns in an Iterative Erasure and Error Decoding Algorithm for Non-binary Block Codes

This letter considers an iterative decoding algorithm for non-binary linear block codes in which erasure and error decoding is performed for input words given by the sums of a hard-decision received sequence and given test patterns. We have proposed a new selection method of test patterns for the iterative decoding algorithm. Simulation results have shown that the decoding algorithm with test patterns by the proposed selection method provides better error performance than a conventional iterative decoding algorithm with the same number of the error and erasure decoding iterations over an additive white Gaussian noise channel using binary phase-shift keying modulation.

Differential space-time modulation over frequency-selective channels

We present herein a new differential space-time-frequency (DSTF) modulation scheme for systems that are equipped with an arbitrary number of transmit antennas and operate in frequency-selective channels. The proposed DSTF modulator consists of a concatenating spectral encoder and differential encoder that offer full spatio-spectral diversity and significant coding gain. A unitary structure is imposed on the differential encoder to admit linear, decoupled maximum likelihood (ML) detection in space and time. Optimum criteria based on pairwise error probability analysis are developed for spectral encoder design. We introduce a class of spectral codes, namely, linear constellation decimation (LCD) codes, which are nonbinary block codes obtained by decimating a phase-shift-keying (PSK) constellation with a group of decimation factors that are co-prime with the constellation size. Since LCD codes encode across a minimally necessary set of subchannels for full diversity, they incur modest decoding complexity among all full-diversity codes. Numerical results are presented to illustrate the performance of the proposed DSTF modulation and coding scheme, which compares favorably with several existing differential space-time schemes in frequency-selective channels.

Construction and performance of algebraic–geometric codes over AWGN and fading channels

Algebraic geometry is a branch of mathematics with many applications such as statistics, computer science and economics. It can also be applied to coding theory to construct nonbinary block codes from curves with impressive parameters e.g. large code lengths, high code rates and large minimum distances. The design and construction of algebraic–geometric codes is described, and the performance evaluated over the additive white gaussian noise (AWGN) and Rayleigh fading channels and make comparisons with similar Reed–Solomon codes. Simulation results show that algebraic–geometric codes constructed from hermitian curves (hermitian codes) outperform Reed–Solomon codes of similar code rate and defined over the same finite field and show that hermitian codes would be suitable for a mobile radio environment.

Diagonal Block Space-Time Code Design for Diversity and Coding Advantage Over Flat Fading Channels

The potential promised by multiple transmit antennas has raised considerable interest in space-time coding for wireless communications. In this paper, we propose a systematic approach for designing space-time trellis codes over flat fading channels with full antenna diversity and good coding advantage. It is suitable for an arbitrary number of transmit antennas with arbitrary signal constellations. The key to this approach is to separate the traditional space-time trellis code design into two parts. It first encodes the information symbols using a one-dimensional (M,1) nonbinary block code, with M being the number of transmit antennas, and then transmits the coded symbols diagonally across the space-time grid. We show that regardless of channel time-selectivity, this new class of space-time codes always achieves a transmit diversity of order M with a minimum number of trellis states and a coding advantage equal to the minimum product distance of the employed block code. Traditional delay diversity codes can be viewed as a special case of this coding scheme in which the repetition block code is employed. To maximize the coding advantage, we introduce an optimal construction of the nonbinary block code for a given modulation scheme. In particular, an efficient suboptimal solution for multilevel phase-shift-keying (PSK) modulation is proposed. Some code examples with 2-6 bits/s/Hz and two to six transmit antennas are provided, and they demonstrate excellent performance via computer simulations. Although it is proposed for flat fading channels, this coding scheme can be easily extended to frequency-selective fading channels.

On combining Chase-2 and GMD decoding algorithms for nonbinary block codes

In this letter, a class of algorithms that combines Chase-2 and GMD (generalized minimum distance) decoding algorithms is presented for nonbinary block codes. This approach provides additional trade-offs between error performance and decoding complexity. Reduced-complexity versions of the algorithms with practical interests are then provided and simulated.

On turbo decoding of nonbinary codes

Symbol-by-symbol maximum a posteriori (MAP) decoding algorithms for nonbinary block and convolutional codes over an extension field GF(p/sup a/) are presented. Equivalent MAP decoding rules employing the dual code are given which are computationally more efficient for high-rate codes. It is shown that these algorithms meet all requirements needed for iterative decoding as the output of the decoder can be split into three independent estimates: soft channel value, a priori term and extrinsic value. The discussed algorithms are then applied to a parallel concatenated coding scheme with nonbinary component codes in conjunction with orthogonal signaling.

Generating Series for Error Statistics of Block Codes on Channels with Memory

We present an analytical method for evaluating the performance of non-interleaved forward error correcting codes on channels that exhibit statistical dependence in the occurrence of errors. We consider a model for such a channel based on the probabilistic function of a Markov chain, also known as finite state channel (FSC) model. The main idea is to apply combinatorial methods to derive simple expressions for the probability of the number of symbol errors produced by the channel in terms of a coefficient in a formal power series. These methods are used to derive expressions for the codeword error probability of various practical coding schemes, including nonbinary block codes and the concatenation of two block codes. The general expressions are specialized for a Gilbert-Elliott channel with known model parameters, and numerical results are derived.

On the decoder error probability of block codes

By using coding and combinational techniques, an explicit formula is derived which enumerates the complete weight distribution of decodable words of block codes using partially known weight distributions. Also, an approximation formula for nonbinary block codes is obtained. These results give exact and approximate expressions for the decoder error probability P/sub E/(u) of block codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new upper bound on nonbinary block codes

An upper bound on the information rate of nonbinary block codes is derived via linear programming in the nonbinary Johnson scheme. Combined with the general cross-section method this new result gives the best presently known asymptotic upper bound on nonbinary block codes.

Error correcting codes for fast frequency‐hopped MFSK spread‐spectrum satellite communications under worst case jamming

AbstractSpread‐spectrum techniques such as frequency hopping (FH) have been used to reduce the vulnerability of satellite communications to jamming. In such a system, error correcting (EC) codes play an essential role. Thus a knowledge of the performance of various EC codes is necessary in order to choose an EC code and related system parameters in the design of an anti‐jam system, this paper examines the performance of various types of error correcting codes under worst case partial band noise jamming and worst case multitone jamming using fast frequency‐hopped, non‐coherent M‐ary frequency‐shift keying (NCMFSK) with optimum diversity. A comprehensive study including convolutional codes, binary and non‐binary block codes and concatenated codes has been conducted. This paper presents bit error rate (BER) performances of various error correcting codes. New candidate codes with superior anti‐jam performance are identified.

Improved Sequential Signaling and Decision Techniques for Nonbinary Block Codes

A situation is postulated whereby the goal is to maximize the average number of bits per second reliably communicated by nonbinary block codes over a channel whose conditions are slowly varying in a largely unpredictable fashion. A feedback channel limited to block retransmission requests is assumed. It is demonstrated that repetition usually is not the best signaling technique for sequential decision making under the postulated situation. Examples of improved sequential signaling techniques are given for multilevel amplitude modulation and multiple phase modulation.