Infinite families of near MDS codes holding $t$-designs

TL;DR

This paper constructs infinite families of near MDS codes over finite fields that hold infinite families of 2- and 3-designs, resolving a 70-year-old open problem in coding theory.

Contribution

It introduces the first known infinite families of near MDS codes holding infinite families of t-designs for t≥2, over specific finite fields.

Findings

Constructed infinite families of near MDS codes over GF(3^s) and GF(2^{2s})

These codes hold infinite families of 3-designs and 2-designs respectively

Subfield subcodes are shown to be dimension- or distance-optimal

Abstract

An $[n, k, n-k+1]$ linear code is called an MDS code. An $[n, k, n-k]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. This ternary code holds $4$-designs, and its extended code holds a Steiner system $S(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding $t$-designs were discovered and many infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of $t$-designs for $t\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $\mathrm{GF}(3^s)$ holding an infinite family of $3$-designs and an infinite family of near MDS codes over $\mathrm{GF}(2^{2s})$ holding an infinite family of $2$-designs. The subfield subcodes of these two families are also studied, and are shown to be dimension-optimal or distance-optimal.