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

Abstract

In this paper we consider a family $\mathcal{F}$ of $16$-dimensional ${\mathbb{F}}_q$-linear rank metric codes in ${\mathbb{F}}_q^{8\times 8}$, 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 Csajbók, 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.

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