Reed-Solomon Subcodes with Nontrivial Traces: Distance Properties and Soft-Decision Decoding

TL;DR

This paper introduces subcodes of Reed-Solomon codes with non-trivial binary traces, analyzing their distance properties and proposing low-complexity decoding algorithms, aiming to improve soft-decision decoding performance.

Contribution

It constructs and studies RS subcodes with non-trivial binary traces, deriving bounds and proposing efficient decoding methods as alternatives to traditional RS codes.

Findings

Subcodes meet derived rate and distance bounds.

Proposed decoders outperform comparable RS codes.

Subcodes are viable for practical applications.

Abstract

Reed-Solomon (RS) codes over GF$(2^m)$ have traditionally been the most popular non-binary codes in almost all practical applications. The distance properties of RS codes result in excellent performance under hard-decision bounded-distance decoding. However, efficient and implementable soft decoding for high-rate (about 0.9) RS codes over large fields (GF(256), say) continues to remain a subject of research with a promise of further coding gains. In this work, our objective is to propose and investigate $2^m$-ary codes with non-trivial binary trace codes as an alternative to RS codes. We derive bounds on the rate of a $2^m$-ary code with a non-trivial binary trace code. Then we construct certain subcodes of RS codes over GF($2^m$) that have a non-trivial binary trace with distances and rates meeting the derived bounds. The properties of these subcodes are studied and low-complexity hard-decision and soft-decision decoders are proposed. The decoders are analyzed, and their performance is compared with that of comparable RS codes. Our results suggest that these subcodes of RS codes could be viable alternatives for RS codes in applications.