On a family of linear MRD codes with parameters $[8\times8,16,7]_q$

TL;DR

This paper investigates a specific family of 16-dimensional linear rank metric codes in 8x8 matrices over finite fields, identifying which are MRD for large odd q and showing their novelty compared to existing codes.

Contribution

It precisely characterizes MRD codes within a family derived from a polynomial, demonstrating their uniqueness among known MRD codes for large odd q.

Findings

Identifies exactly which codes in the family are MRD for large odd q.

Shows these MRD codes are not equivalent to previously known MRD codes.

Abstract

In this paper we consider a family $\mathcal{F}$ of $16$-dimensional $\mathbb{F}_q$-linear rank metric codes in $\mathbb{F}_q^{8\times8}$, arising from the polynomial $x^{q^s}+\delta x^{q^{4+s}}\in\mathbb{F}_{q^8}[x]$. Examples of MRD codes in $\mathcal{F}$ have been provided by Csajb\'ok, Marino, Polverino and Zanella (2018). For any large enough odd $q$, we determine exactly which codes in $\mathcal{F}$ are MRD. We also show that the MRD codes in $\mathcal{F}$ are not equivalent to any other MRD codes known so far.