Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights

TL;DR

This paper introduces a tree-based method for constructing LDPC codes with high pseudocodeword weights, improving decoding performance and code properties, including for p-ary codes, by employing permutation and Latin square-based connections.

Contribution

The paper presents a novel tree-based construction method for LDPC codes that achieves high pseudocodeword weights and introduces new classes of codes, extending finite geometry LDPC codes to p-ary cases.

Findings

Constructed LDPC codes with pseudocodeword weight close to minimum distance.

Codes perform well with iterative decoding.

Extension to p-ary LDPC codes improves rates and distances.

Abstract

We present a tree-based construction of LDPC codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a $d$-regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal Latin squares to close the tree. Methods are presented for degrees $d=p^s$ and $d = p^s+1$, for $p$ a prime. One class corresponds to the well-known finite-geometry and finite generalized quadrangle LDPC codes; the other codes presented are new. We also present some bounds on pseudocodeword weight for $p$-ary LDPC codes. Treating these codes as $p$-ary LDPC codes rather than binary LDPC codes improves their rates, minimum distances, and pseudocodeword weights, thereby giving a new importance to the finite geometry LDPC codes where $p > 2$.