Product Construction of Affine Codes

Binary matrix codes with restricted row and column weights are a desirable method of coded modulation for power line communication. In this work, we construct such matrix codes that are obtained as products of affine codes - cosets of binary linear codes. Additionally, the constructions have the property that they are systematic. Subsequently, we generalize our construction to irregular product of affine codes, where the component codes are affine codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic

Irregular Product Codes

We introduce irregular product codes, a class of codes where each codeword is represented by a matrix and the entries in each row (column) of the matrix come from a component row (column) code. As opposed to standard product codes, we do not require that all component row codes nor all component column codes be the same. Relaxing this requirement can provide some additional attractive features such as allowing some regions of the codeword to be more error-resilient, providing a more refined spectrum of rates for finite lengths, and improved performance for some of these rates. We study these codes over erasure channels and prove that for any 0 < ε < 1, for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has asymptotic rate 1 - ε and can decode on erasure channels having erasure probability <; ε (and having alphabet size equal to the alphabet size of the component MDS codes)

Density Evolution for Deterministic Generalized Product Codes with Higher-Order Modulation

Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case where transmission takes place over parallel BECs with different erasure probabilities. We use this model to predict the code performance in a coded modulation setup with higher-order signal constellations. We also discuss the design of the bit mapper that determines the allocation of the coded bits to the modulation bits of the signal constellation.Comment: invited and accepted paper for the special session "Recent Advances in Coding for Higher Order Modulation" at the International Symposium on Turbo Codes & Iterative Information Processing, Brest, France, 201

Generalized Spatially-Coupled Product-Like Codes Using Zipper Codes With Irregular Degree

Zipper codes with irregular variable degree are studied. Two new interleaver maps -- chevron and half-chevron -- are described. Simulation results with shortened double-error-correcting Bose--Chaudhuri--Hocquenghem constituent codes show that zipper codes with chevron and half-chevron interleaver maps outperform staircase codes when the rate is below 0.86 and 0.91, respectively, at $10^{-8}$ output bit error rate operating point. In the miscorrection-free decoding scheme, both zipper codes with chevron and half-chevron interleaver maps outperform staircase codes. However, constituent decoder miscorrections induce additional performance gaps.Comment: 6 pages, 11 figures, paper accepted for the GLOBECOM 2023 Workshop on Channel Coding Beyond 5

Low-complexity bound on irregular LDPC belief-propagation decoding thresholds using a Gaussian approximation

Since irregular low-density parity-check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so-called \u2018threshold phenomenon\u2019, this Letter investigates a low-complexity upper bound on belief-propagation decoding thresholds for this class of codes on memoryless binary input additive white Gaussian noise channels, with sum-product decoding. A simplified analysis of the belief-propagation decoding algorithm is used, i.e. consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes

Useful Mathematical Tools for Capacity Approaching Codes Design

Focus of this letter is the oldest class of codes that can approach the Shannon limit quite closely, i.e., lowdensity parity-check (LDPC) codes, and two mathematical tools that can make their design an easier job under appropriate assumptions. In particular, we present a simple algorithmic method to estimate the threshold for regular and irregular LDPC codes on memoryless binary-input continuous-output AWGN channels with sum-product decoding, and, to determine how close are the obtained thresholds to the theoretical maximum, i.e., to the Shannon limit, we give a simple and invertible expression of the AWGN channel capacity in the binary input - soft output case. For these codes, the thresholds are defined as the maximum noise level such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We assume a Gaussian approximation for message densities under density evolution, a widely used simplification of the decoding algorithm