New infinite families of near MDS codes holding $t$-designs

TL;DR

This paper introduces new infinite families of near MDS codes that support $t$-designs, expanding the known classes and providing codes with different parameters, along with optimal locally recoverable codes.

Contribution

It constructs new infinite families of NMDS codes supporting $t$-designs with distinct parameters and determines their weight enumerators.

Findings

New infinite families of NMDS codes supporting $2$- and $3$-designs

Determination of weight enumerators for these NMDS codes

Derivation of optimal locally recoverable codes from the NMDS codes

Abstract

In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $t$-designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes.