Properties of subspace subcodes of Gabidulin codes (Q935617)

This paper extends the results previously obtained and presented by \textit{E. M. Gabidulin, P. Loindreau} [in: 2005 IEEE International Symposium on Information Theory, 121--123 (2005)] and shows that in some cases questions remaining open for subspace subcodes of Reed-Solomon codes in Hamming metric can be answered for Gabidulin codes in rank metric: When the length of the parent Gabidulin code is equal to the degree of the alphabet field, there exists a rank-preserving isomorphism between the subspace subcode and a Gabidulin code with smaller parameters. In that case. the authors design a systematic procedure encoding all the possible information as well as a decoding algorithm correcting up to the capability of the subspace subcode. Then, they generalise the results to the direct sum of subspace subcodes and show that the number of decodable error-patterns is larger than what is theoreticalIy possible for a code with the same parameters, but without this additional structure. Finally they prove that subfield subcodes of Gabidulin codes can be seen, modulo the action of the generallinear group, as the direct sum of Gabidulin codes with smaller parameters.