Optimal Anticodes, Diameter Perfect Codes, Chains and Weights

Abstract: Let $P$ be a partial order on $[n] = {1,2,\ldots,n}$, $\mathbb{F}{q}^n$ be the linear space of $n$-tuples over a finite field $\mathbb{F}{q}$ and $w$ be a weight on $\mathbb{F}{q}$. In this paper, we consider metrics on $\mathbb{F}{q}^n$ induced by chain orders $P$ over $[n]$ and weights $w$ over $\mathbb{F}_q$, and we determine the cardinality of all optimal anticodes and completely classify them. Moreover, we determine all diameter perfect codes for a set of relevant instances on the aforementioned metric spaces.