On the Pseudocodeword Redundancy of Binary Linear Codes

TL;DR

This paper investigates the pseudocodeword redundancy of binary linear codes, establishing conditions for finiteness, providing bounds for specific code families, and computing redundancies for small and cyclic codes.

Contribution

It introduces the concept of pseudocodeword redundancy, shows most codes lack finite redundancy, and offers bounds and exact values for various code classes.

Findings

Most codes do not have finite pseudocodeword redundancy.

Upper bounds are established for codes based on designs.

Redundancies for all codes up to length 9 are computed.

Abstract

The AWGNC, BSC, and max-fractional pseudocodeword redundancies of a binary linear code are defined to be 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. It is shown that most codes do not have a finite pseudocodeword redundancy. Also, upper bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs, are provided. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Furthermore, 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.