Pseudocodewords from Bethe Permanents

TL;DR

This paper proves a stronger version of a conjecture linking Bethe permanents to pseudocodewords and explores their properties, enhancing understanding of their role in coding theory.

Contribution

It establishes a stronger proof for the conjecture in key cases and analyzes the properties of pseudocodewords derived from Bethe permanents.

Findings

Proved the conjecture for important cases

Identified properties of permanents of block matrices

Analyzed the structure of Bethe permanent-based pseudocodewords

Abstract

It was recently conjectured that a vector with components equal to the Bethe permanent of certain submatrices of a parity-check matrix is a pseudocodeword. In this paper we prove a stronger version of this conjecture for some important cases and investigate the families of pseudocodewords obtained by using the Bethe permanent. We also highlight some interesting properties of the permanent of block matrices and their effects on pseudocodewords.