On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes

TL;DR

This paper introduces a new low-rate TLDPC code ensemble designed to meet a necessary condition for linear minimum distance, achieving near-Shannon-limit thresholds with low complexity and fast convergence.

Contribution

It proposes a novel low-rate TLDPC code ensemble satisfying a derived necessary condition for linear minimum distance, with favorable decoding performance.

Findings

Ensemble's iterative threshold is close to Shannon limit

Codes have low decoding complexity and fast convergence

Satisfies a necessary condition for linear minimum distance

Abstract

This paper addresses the issue of design of low-rate sparse-graph codes with linear minimum distance in the blocklength. First, we define a necessary condition which needs to be satisfied when the linear minimum distance is to be ensured. The condition is formulated in terms of degree-1 and degree-2 variable nodes and of low-weight codewords of the underlying code, and it generalizies results known for turbo codes [8] and LDPC codes. Then, we present a new ensemble of low-rate codes, which itself is a subclass of TLDPC codes [4], [5], and which is designed under this necessary condition. The asymptotic analysis of the ensemble shows that its iterative threshold is situated close to the Shannon limit. In addition to the linear minimum distance property, it has a simple structure and enjoys a low decoding complexity and a fast convergence.