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.

翻译：在“无限族近MDS码持有$t$-设计，IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428”中，Ding和Tang取得了突破性进展，首次构造了两个持有$2$-设计或$3$-设计的NMDS码无穷族。截至目前，文献中已知的持有$t$-设计的NMDS码无穷族仅有少数几个。本文旨在构造持有$t$-设计的NMDS码新无穷族。我们确定了NMDS码的重量计数器，并证明了这些NMDS码持有$2$-设计或$3$-设计。与已知的由NMDS码导出的$t$-设计相比，我们的设计具有不同的参数。此外，通过该NMDS码还推导出了若干无穷族最优局部修复码。