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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

Peter Vandendriessche 1,

1. 

Boudewijn Hapkenstraat 5, 8820 Torhout, Belgium

Received  December 2009 Revised  May 2010 Published  August 2010

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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