Optimal Anticodes, Diameter Perfect Codes, Chains and Weights

TL;DR

This paper studies metrics on finite field vector spaces induced by chain orders and weights, classifies optimal anticodes, and identifies diameter perfect codes for specific cases, advancing coding theory in structured metric spaces.

Contribution

It provides a complete classification of optimal anticodes and determines diameter perfect codes in metrics induced by chain orders and weights.

Findings

Classified all optimal anticodes in the considered metric spaces.

Determined all diameter perfect codes for relevant instances.

Extended understanding of code structures under chain order-induced metrics.

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.