Symmetric matrices and codes correcting rank errors beyond the $\lfloor (d-1)/2\rfloor$ bound

Abstract

Rank codes can be described either as matrix codes over the base field $F_q$ or as vector codes over the extension field $F_{q^n}$. For any matrix code, there exists a corresponding vector codes, and vice versa. We investigate matrix codes containing a linear subcode of symmetric matrices. The corresponding vector codes contain a linear subspace of so-called symmetric vectors. It is shown that such vector codes are generated by self-orthogonal bases of the field $F_{q^n}.$ If code distance is equal to $d$, than such codes can correct not only all the errors of rank up to $\lfloor (d-1)/2\rfloor$ but also many symmetric errors of rank beyond this bound.