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 $(n-k-1)$-$(n,n-k,λ)$ design. We first determine the parameters of the dual code $\mathcal{C}^{\bot}$. We also characterize the design an