Let $k\leq n \leq p$, where $p$ is prime. I am trying to proof that linear code $K = L^{\overline{0,n-1}}_{\mathbb{F}_p}((x-e)^k)$ is MDS code.
$L^{\overline{0,n-1}}_{\mathbb{F}_p}((x-e)^k)$ is the set of LRS' segments of length n with characteristic polynomial $(x-e)^k$.
It's easy to see that dim K = k.
At first I used Singleton's bound, so d(K) $\leq$ n-k+1. Also we can say that $e^{[0]}(\overline{0,n-1}),\dots,e^{[k-1]}(\overline{0,n-1}) - $ binomial basis of K. Let d(x,y) be a Hamming distance between words x and y. Define $\overline{0}$ as a "null word". So $$d(e^{[0]}(\overline{0,n-1}),\overline{0}) > \dots > d(e^{[k-1]}(\overline{0,n-1}),\overline{0}) = n-k+1 $$ I think it remains to prove that for all words "y" in K $$ d(y,\overline{0}) \geq n-k+1 $$