Properties of subspace subcodes of optimum codes in rank metric

Maximum rank distance codes denoted MRD-codes are the equivalent in rank metric of MDS-codes. Given any integer $q$ power of a prime and any integer $n$ there is a family of MRD-codes of length $n$ over $\FF{q^n}$ having polynomial-time decoding algorithms. These codes can be seen as the analogs of Reed-Solomon codes (hereafter denoted RS-codes) for rank metric. In this paper their subspace subcodes are characterized. It is shown that hey are equivalent to MRD-codes constructed in the same way but with smaller parameters. A specific polynomial-time decoding algorithm is designed. Moreover, it is shown that the direct sum of subspace subcodes is equivalent to the direct product of MRD-codes with smaller parameters. This implies that the decoding procedure can correct errors of higher rank than the error-correcting capability. Finally it is shown that, for given parameters, subfield subcodes are completely characterized by elements of the general linear group ${GL}_n(\FF{q})$ of non-singular $q$-ary matrices of size $n$.