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

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.