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arXiv:2309.14917 · IEEE Transactions on Communications

Rate-Compatible LDPC Codes Based on Primitive Polynomials and Golomb Rulers

Massimo Battaglioni, Marco Baldi, Franco Chiaraluce, Giovanni Cancellieri

Published Jun 17, 2024• 9 citations
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Abstract

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.

DOI
10.1109/tcomm.2024.3415597

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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.
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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 propose an algebraic approach to design extremely low rate low-density parity-check (LDPC) codes by employing the primitive polynomials. We first investigate the requirements for the polynomials to construct 4-cycle free LDPC codes and then propose an algebraic approach to design the parity check matrix. To achieve a better decoding performance in the extremely low rate region, we propose an extension method to construct more check nodes and virtual variable nodes to obtain more constraints. Our designed LDPC codes have simple encoding and simple decoding features. Simulation results show that our proposed rate-0.03 LDPC code can achieve 2.2 dB gain, compared to the LDPC code designed by the progressive edge growth algorithm with the optimized edge degree distributions described in [7].