Exploration of AWGNC and BSC Pseudocodeword Redundancy

TL;DR

This paper investigates the minimal parity-check matrix size needed for certain pseudoweight properties in codes, providing new theoretical results and computational data for small and cyclic codes.

Contribution

It offers new bounds and exact values for pseudocodeword redundancies of various codes, including small and cyclic codes, and analyzes the sharpness of eigenvalue bounds.

Findings

Pseudocodeword redundancies computed for all codes up to length 9.

Results on the sharpness of eigenvalue bounds for cyclic codes up to length 250.

New bounds and exact values for pseudocodeword redundancies.

Abstract

The AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code is defined as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. This paper provides new results on the AWGNC, BSC, and max-fractional pseudocodeword redundancies of codes. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Also, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.