On Almost MDS codes holding $(n-k-1)$-designs and an open problem on Witt design

Let $\mathcal{C}$ be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an $(n-k-1)$-$(n,n-k,\lambda )$ design. We first determine the parameters of the dual code ${\mathcal{C}}^{\mathrm{\perp }}$. We also characterize the design and its complementary design supported by the minimum weight codewords of ${\mathcal{C}}^{\mathrm{\perp }}$. In particular, we present explicit formulas for the weight enumerators of all subset codes of ${\mathcal{C}}^{\mathrm{\perp }}$ with length at least $k+1$. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter $\lambda$, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.