A New Formula for the Minimum Distance of an Expander Code

TL;DR

This paper introduces a new formula for calculating the minimum distance of expander codes, enhancing understanding of their error-correcting capabilities and providing a new proof for existing bounds.

Contribution

It presents a novel formula for the minimum distance of expander codes and offers a new proof for a known lower bound based on bipartite expander graphs.

Findings

New formula for minimum distance of expander codes

Alternative proof for lower bound of minimum distance

Improved theoretical understanding of expander code properties

Abstract

An expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. We provide a new formula for the minimum distance of such codes. We also provide a new proof of the result that $2(1-\varepsilon) \gamma n$ is a lower bound of the minimum distance of the expander code given by a $(m,n,d,\gamma,1-\varepsilon)$ expander bipartite graph.