LDPC codes associated with linear representations of geometries

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August 2010, 4(3): 405-417. doi: 10.3934/amc.2010.4.405

LDPC codes associated with linear representations of geometries

1.	Boudewijn Hapkenstraat 5, 8820 Torhout, Belgium

Received December 2009 Revised May 2010 Published August 2010

Abstract
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We look at low density parity check codes over a finite field $\mathbb K$ associated with finite geometries $T$₂^*$(\mathcal K)$, where $\mathcal K$ is any subset of PG$(2,q)$, with $q=p$^h, $p$≠char$\mathbb K$. This includes the geometry $LU(3,q)$^D, the generalized quadrangle $T$₂^*$(\mathcal K)$ with $\mathcal K$ a hyperoval, the affine space AG$(3,q)$ and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of $\mathcal K$.

Keywords: minimum distance, linear representation, LDPC codes, finite geometry, linear codes, dimension..

Mathematics Subject Classification: Primary: 51E15, 51E20, 94B0.

Citation: Peter Vandendriessche. LDPC codes associated with linear representations of geometries. Advances in Mathematics of Communications, 2010, 4 (3) : 405-417. doi: 10.3934/amc.2010.4.405

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