Optimal Anticodes, Diameter Perfect Codes, Chains and Weights

Abstract

Let $P$ be a partial order on $[n]=\{1,2,\dots ,n\}$, ${\mathbb{F}}_q^n$ 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.

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