Distance properties of expander codes

TL;DR

This paper investigates the minimum distance of bipartite graph-based codes, demonstrating that certain expander codes are asymptotically good and can surpass traditional bounds, with implications for code design.

Contribution

It provides new insights into the distance properties of expander codes, including asymptotic goodness and surpassing the product bound across all code rates.

Findings

Random ensemble codes have calculable weight spectrum and minimum distance.

Codes with vertex codes of minimum distance ≥ 3 are asymptotically good.

Constructive expander codes can exceed the product bound for all code rates.

Abstract

We study the minimum distance of codes defined on bipartite graphs. Weight spectrum and the minimum distance of a random ensemble of such codes are computed. It is shown that if the vertex codes have minimum distance $\ge 3$, the overall code is asymptotically good, and sometimes meets the Gilbert-Varshamov bound. Constructive families of expander codes are presented whose minimum distance asymptotically exceeds the product bound for all code rates between 0 and 1.