Upper and Lower Bounds on the Minimum Distance of Expander Codes

TL;DR

This paper investigates the minimum distance of expander codes over GF(q), deriving new upper and lower bounds, with some bounds approaching the Varshamov-Gilbert limit, especially for LDPC codes with Reed--Solomon components.

Contribution

It introduces a new upper bound for expander codes' minimum distance and provides lower bounds for specific families, including LDPC codes with Reed--Solomon constituents.

Findings

New upper bound on expander codes' minimum distance

Lower bounds for certain expander code families

LDPC codes with Reed--Solomon components have bounds close to VG bound

Abstract

The minimum distance of expander codes over GF(q) is studied. A new upper bound on the minimum distance of expander codes is derived. The bound is shown to lie under the Varshamov-Gilbert (VG) bound while q >= 32. Lower bounds on the minimum distance of some families of expander codes are obtained. A lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed--Solomon constituent code over GF(q) is obtained. The bound is shown to be very close to the VG bound and to lie above the upper bound for expander codes.