On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

TL;DR

This paper introduces an improved union bound for LP decoding of high-density parity-check codes using a second-order Bonferroni inequality and Prim's algorithm, enhancing performance estimation accuracy.

Contribution

It proposes a novel bound based on second-order inequalities, utilizing fundamental cone generators, and analyzes generator density effects on the bound's improvement.

Findings

The new bound outperforms the conventional LP union bound.

Generator density significantly affects the bound's tightness.

Complete pseudo-weight distribution for BCH[31,21,5] code is provided.

Abstract

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code.