On a family of linear MRD codes with parameters [8 x 8, 16, 7](q)

In this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that 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 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 F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far.