Near MDS codes from oval polynomials

TL;DR

This paper constructs multiple infinite families of near MDS codes over finite fields using oval polynomials, expanding the known classes of such codes with specific parameters.

Contribution

It introduces new infinite families of near MDS codes with specific parameters derived from oval polynomials, enriching the finite geometry and coding theory literature.

Findings

Seven families of near MDS codes with parameters [2^m+1, 3, 2^m-2] over GF(2^m)

Seven families of near MDS codes with parameters [2^m+2, 3, 2^m-1] over GF(2^m)

Nine families of optimal near MDS codes with parameters [2^m+3, 3, 2^m] over GF(2^m)

Abstract

A linear code with parameters of the form $[n, k, n-k+1]$ is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form $[n, k, n-k]$ is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable. Near MDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. In this paper, seven infinite families of $[2^m+1, 3, 2^m-2]$ near MDS codes over $\gf(2^m)$ and seven infinite families of $[2^m+2, 3, 2^m-1]$ near MDS codes over $\gf(2^m)$ are constructed with special oval polynomials for odd $m$. In addition, nine infinite families of optimal $[2^m+3, 3, 2^m]$ near MDS codes over $\gf(2^m)$ are constructed with oval polynomials in general.