Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

TL;DR

This paper introduces Primitive Rate-Compatible LDPC codes derived from primitive polynomials and Golomb rulers, offering simple encoding, adjustable rates, and promising error correction performance.

Contribution

It presents a novel class of LDPC codes with simple encoders, rate flexibility via puncturing, and a new design criterion involving Golomb rulers.

Findings

Codes exhibit good minimum distance properties

Encoding and decoding complexities are manageable

Error rate performance is promising under iterative decoding

Abstract

We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes characterized by very simple encoders. The design of these codes starts from simplex codes, which are defined by parity-check matrices having a straightforward 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 their code rates. 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 the 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 their error rate performance under iterative decoding.