Properties of subspace subcodes of Gabidulin codes

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

Properties of subspace subcodes of Gabidulin codes

Ernst M. Gabidulin ^1, and Pierre Loidreau ^2,

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

Abstract
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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 $n$.

Keywords: Gabidulin codes., Coding theory, subcodes structure.

Mathematics Subject Classification: Primary: 11T71, 11T55; Secondary: 14G5.

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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