Improved Nearly-MDS Expander Codes

TL;DR

This paper introduces a new construction of expander codes that are close to the Singleton bound, efficiently encodable and decodable, and capable of replacing MDS codes in concatenated schemes to approach channel capacity.

Contribution

It presents a novel expander code construction with near-Singleton bound properties, linear-time encoding/decoding, and improved alphabet size efficiency over previous methods.

Findings

Codes approach the Singleton bound.

Encoding and decoding are linear time.

Codes can replace MDS codes in concatenated schemes.

Abstract

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size.