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.