Extending two families of maximum rank distance codes

TL;DR

This paper introduces a broad new family of maximum rank distance codes over finite fields, extending previous constructions and establishing their novelty and inequivalence to known codes for certain parameters.

Contribution

It generalizes existing rank-metric codes by providing a large family of new maximum rank distance codes that are inequivalent to prior known constructions for t ≥ 5.

Findings

The codes are $ ext{F}_{q^{2t}}$-linear of dimension 2.

They are proven to be maximum rank distance codes.

For t ≥ 5, the codes are inequivalent to all previously known codes.

Abstract

In this paper we provide a large family of rank-metric codes, which contains properly the codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are $\mathbb{F}_{q^{2t}}$-linear of dimension $2$ in the space of linearized polynomials over $\mathbb{F}_{q^{2t}}$, where $t$ is any integer greater than $2$, and we prove that they are maximum rank distance codes. For $t\ge 5$, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.