The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes

TL;DR

This paper investigates subfield-subcodes of shortened generalized Reed-Solomon codes, presenting an algorithm to find many with large dimension and providing new code examples.

Contribution

It introduces a method to identify high-dimension subfield-subcodes of GRS codes and offers new code constructions not previously documented.

Findings

Many subfield-subcodes with large dimension are produced by a specific class of GRS codes.

An algorithm for searching and identifying these codes is developed.

New codes with improved parameters are listed and analyzed.

Abstract

Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is static, the appropriate length can be obtained by an RS code by shortening or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS codes, whose subfield-subcodes are extensively studied. In this paper we show that a particular class of GRS codes produces many subfield-subcodes with large dimension. An algorithm for searching through the codes is presented as well as a list of new codes obtained from this method.