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 2n-dimensional \({\mathbb {F}}_q\)-linear rank metric codes in \({\mathbb {F}}_q^{n\times n}\) arising from polynomials of the form \(x^{q^s}+\delta x^{q^{\frac{n}{2}+s}}\in {\mathbb {F}}_{q^n}[x]\). The family \(\mathcal {F}\) was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that \(\mathcal {F}\) contains MRD codes for \(n=8\), and other subsequent partial results have been provided in the literature towards the classification of MRD codes in \(\mathcal {F}\) for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case \(n=8\), providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional \({\mathbb {F}}_q\)-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in \(\mathcal {F}\) are not equivalent to any other MRD codes known so far.