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International Conference on Communications

Rate-Adaptive LDPC Codes Obtained from Simplex Codes

Massimo Battaglioni, Marco Baldi, Franco Chiaraluce, Giovanni Cancellieri

Published May 28, 2023• 6 citations
Publisher

Abstract

In this paper we show that, when a binary primitive polynomial can be associated to a sparse Golomb ruler, the simplex code obtained by taking it as the code parity-check polynomial exhibits good distance properties and performance. We define some conditions under which the obtained codes are also Low-Density Parity-Check (LDPC) codes, and can hence be decoded through efficient iterative algorithms. We perform code puncturing, leading to a family of rate-adaptive codes, and we predict some of their structural properties in terms of minimum distance and weight distribution. We show that, in addition to having some useful properties, these codes achieve good performance in terms of error rate under LDPC decoding.

DOI
10.1109/icc45041.2023.10279448

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In this paper we show that, when a binary primitive polynomial can be associated to a sparse Golomb ruler, the simplex code obtained by taking it as the code parity-check polynomial exhibits good distance properties and performance. We define some conditions under which the obtained codes are also Low-Density Parity-Check (LDPC) codes, and can hence be decoded through efficient iterative algorithms. We perform code puncturing, leading to a family of rate-adaptive codes, and we predict some of their structural properties in terms of minimum distance and weight distribution. We show that, in addition to having some useful properties, these codes achieve good performance in terms of error rate under LDPC decoding.
We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes. The design of these codes starts from simplex codes, defined by parity-check matrices having a simple form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of the code rate. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take these new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess the error rate performance of high rate PRC-LDPC codes under iterative decoding.
Digital data transfer can be protected by means of suitable error correcting codes. Among the families of state-of-the-art codes, LDPC (Low Density Parity-Check) codes have received a great deal of attention recently, because of their performance and flexibility of operation, in wireless and mobile radio channels, as well as in cable transmission systems. In this paper, we present a class of rate-adaptive LDPC codes, obtained as properly punctured simplex codes. These codes allow for the use of an efficient soft-decision decoding algorithm, provided that a condition called row-column constraint is satisfied. This condition is tested on small-length codes, and then extended to medium-length codes. The puncturing operations we apply do not influence the satisfaction of the row-column constraint, assuring that a wide range of code rates can be obtained. We can reach code rates remarkably higher than those obtainable by the original simplex code, and the price in terms of minimum distance turns out to be relatively small, leading to interesting trade-offs in the resulting asymptotic coding gain.
In this paper we study a special family of Low-Density Parity-Check (LDPC) codes, called Primitive Rate-Compatible (PRC) LDPC codes, obtained by applying puncturing to some properly chosen simplex codes. We investigate their minimum distance properties and give insights on the design of PRC-LDPC codes for use in scenarios where constraints on hardware and software resources require maintaining a low level of complexity. We numerically assess the error rate performance of these codes in such a setting, showing that they also demonstrate good performance under belief propagation decoding, in addition to low decoding complexity.
Low density parity check (LDPC) codes with iterative decoding based on belief propagation achieve astonishing error performance close to Shannon limit. No algebraic or geometric method for constructing these codes has been reported and they are largely generated by computer search. As a result, encoding of long LDPC codes is in general very complex. This paper presents two classes of high rate LDPC codes whose constructions are based on finite Euclidean and projective geometries, respectively. These classes of codes a.re cyclic and have good constraint parameters and minimum distances. Cyclic structure adows the use of linear feedback shift registers for encoding. These finite geometry LDPC codes achieve very good error performance with either soft-decision iterative decoding based on belief propagation or Gallager's hard-decision bit flipping algorithm. These codes can be punctured or extended to obtain other good LDPC codes. A generalization of these codes is also presented.
Based jointly on idempotents and modular Golomb rulers, we construct a class of nonbinary cyclic low-density parity-check (LDPC) codes. The defining parity-check matrix is a sparse circulant, on which we put two constraints: 1) the characteristic polynomial is an idempotent, 2) the nonzero elements of the first row are located on a modular Golomb ruler. We show that the second constraint forms a necessary and sufficient condition for the Tanner graph to have no cycles of length 4. The minimum distance of the code is proved equal to the column weight of the parity-check matrix plus one. A search algorithm is presented, with which we obtain some high rate codes with large minimum distances. The issue of code equivalence is also discussed. Simulation results show that the obtained codes perform well under iterative decoding.
Traditionally, low-density parity-check (LDPC) codes are constructed randomly and it is not clear whether algebraically constructed codes, such as cyclic codes, can compete with the error correction performances of the random codes. We consider in this paper new cyclic low-density parity-check codes decoded with the sum-product algorithm. The cyclic codes we present have sparse parity-check matrices, good minimum distance and girth 6. Using frnite-length analysis and simulation results we show that the most significant benefit of the new codes, in terms of error correction performance, is the linearly dependent rows in their parity-check matrices. Our results suggest that increasing column weight to improve the minimum distance and the minimum stopping set size of the code proves beneficial at low erasure probabilities but adversely affects performance in channels with high erasure probabilities.