Quasi optimal anticodes: structure and invariants

TL;DR

This paper investigates quasi optimal anticodes in the rank-metric, focusing on their structure, invariants, and dual properties, extending understanding beyond optimal anticodes to codes with minimal maximum rank for given dimensions.

Contribution

It introduces and characterizes quasi optimal anticodes and their duals, providing explicit structural descriptions and invariant computations.

Findings

Explicit structure of dually qOACs

Computed weight distributions and generalized weights

Analyzed associated q-polymatroids

Abstract

It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.