MDS codes over finite fields

TL;DR

This paper investigates the bounds of maximum distance separable (MDS) codes over finite fields, constructs new codes meeting these bounds, and discusses their encoding and decoding efficiency.

Contribution

It proves bounds for MDS codes over finite fields and constructs new codes that achieve these bounds, including for special cases when q is even.

Findings

MDS codes over GF(q) satisfy specific length bounds depending on q.

Constructed new MDS codes for q+1 and q+2 lengths.

Codes have efficient encoding and decoding algorithms.

Abstract

The mds (maximum distance separable) conjecture claims that a nontrivial linear mds $[n,k]$ code over the finite field $GF(q)$ satisfies $n \leq (q + 1)$, except when $q$ is even and $k = 3$ or $k = q- 1$ in which case it satisfies $n \leq (q + 2)$. For given field $GF(q)$ and any given $k$, series of mds $[q+1,k]$ codes are constructed. Any $[n,3]$ mds or $[n,n-3]$ mds code over $GF(q)$ must satisfy $n\leq (q+1)$ for $q$ odd and $n\leq (q+2)$ for $q$ even. For even $q$, mds $[q+2,3]$ and mds $[q+2, q-1]$ codes are constructed over $GF(q)$. The codes constructed have efficient encoding and decoding algorithms.