Linear complete symmetric rank-distance codes

Abstract

An ${\mathbb{F}}_q$-linear code of minimum distance $d$ is called complete if it is not contained in a larger ${\mathbb{F}}_q$-linear code of minimum distance $d$. In this paper, we classify ${\mathbb{F}}_q$-linear complete symmetric rank-distance (CSRD) codes in $M_{3\times 3}({\mathbb{F}}_q)$ up to equivalence. This includes the classification of ${\mathbb{F}}_q$-linear maximum symmetric rank-distance (MSRD) codes in $M_{3\times 3}({\mathbb{F}}_q)$. Our approach is mainly geometric, and our results contribute towards the classification of nets of conics in $\mathrm{PG}(2,q)$.