The Subfield Codes of $[q+1, 2, q]$ MDS Codes

TL;DR

This paper introduces a general construction for $[q+1, 2, q]$ MDS codes over finite fields and explores their subfield codes over smaller fields, resulting in dimension-optimal and nearly optimal codes.

Contribution

It presents a new general construction of $[q+1, 2, q]$ MDS codes and analyzes their subfield codes, producing new optimal and nearly optimal codes over small fields.

Findings

Two families of dimension-optimal codes over $ ext{GF}(p)$ were obtained.

Several families of nearly optimal codes over $ ext{GF}(p)$ were produced.

Open problems related to subfield codes were proposed.

Abstract

Recently, subfield codes of geometric codes over large finite fields $\gf(q)$ with dimension $3$ and $4$ were studied and distance-optimal subfield codes over $\gf(p)$ were obtained, where $q=p^m$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over $\gf(q)$, and then studies the subfield codes over $\gf(p)$ of some of the $[q+1, 2,q]$ MDS codes over $\gf(q)$. Two families of dimension-optimal codes over $\gf(p)$ are obtained, and several families of nearly optimal codes over $\gf(p)$ are produced. Several open problems are also proposed in this paper.