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Properties of subspace subcodes of Gabidulin codes
We investigate properties of subspace subcodes of Gabidulin codes. They are isomorphic to Gabidulin codes with the same minimum rank distance and smaller parameters. We design systematic encoding and decoding algorithms for subspace subcodes. We show that the direct sum of subspace subcodes of Gabidulin codes is isomorphic to the direct product of Gabidulin
codes with smaller parameters. Thanks to this structure there is a great deal of correctable error-patterns whose rank exceeds the error-correcting capability. Finally we show that for particular sets of parameters, subfield subcodes of Gabidulin codes can be uniquely characterised by elements of the general linear group GL$_n(GF(q))$ of non-singular $q$-ary matrices of size $n$.