On the Pseudocodeword Redundancy

TL;DR

This paper investigates the minimal number of parity-check equations needed for a code's pseudoweight to match its minimum Hamming distance, revealing that many codes lack finite pseudocodeword redundancy and providing bounds for certain code families.

Contribution

It introduces the concept of pseudocodeword redundancy for different channels and establishes that most codes do not have finite values, offering bounds for specific code families.

Findings

Most codes do not have finite pseudocodeword redundancy.

Bounds are provided for codes based on designs.

The minimal number of parity-checks for certain pseudoweights is characterized.

Abstract

We define the AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance. We show that most codes do not have a finite pseudocodeword redundancy. We also provide bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs.