# American Institute of Mathematical Sciences

May  2008, 2(2): 147-157. doi: 10.3934/amc.2008.2.147

## Properties of subspace subcodes of Gabidulin codes

 1 Moscow Institute of Physics and Technology, Institutskii per. 9, 141700 Dolgoprudny, Moscow Region, Russian Federation 2 CELAR and IRMAR, Université de Rennes, La Roche Marguerite, BP 57419, 35171 Bruz, France

Received  September 2007 Revised  February 2008 Published  April 2008

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 $n$.
Citation: Ernst M. Gabidulin, Pierre Loidreau. Properties of subspace subcodes of Gabidulin codes. Advances in Mathematics of Communications, 2008, 2 (2) : 147-157. doi: 10.3934/amc.2008.2.147
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