Symmetric matrices and codes correcting rank errors beyond the ⌊(d-1)/2⌋⌊(d-1)/2⌋ bound

Rank codes can be described either as matrix codes over the base field FqFq or as vector codes over the extension field FqnFqn. 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 Fqn.Fqn. If code distance is equal to dd, than such codes can correct not only all the errors of rank up to ⌊(d-1)/2⌋⌊(d-1)/2⌋ but also many symmetric errors of rank beyond this bound.