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

Ziling Heng; Xinran Wang

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arXiv · 2210.05194

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

Ziling Heng·Xinran Wang

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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.

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Ziling Heng, Xinran Wang. 2022-10-11. New infinite families of near MDS codes holding $t$-designs. https://arxiv.org/abs/2210.05194

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