Small weight codewords in the LDPC codes arising from linear representations of geometries

Abstract

AbstractIn this article, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods. First, we present a new lower bound on the minimum distance and we present a number of cases in which this lower bound is sharp. Then we take a closer look at the cases$T_2^*(\Theta)$and$T_2^*(\Theta)^D$with$\Theta$a hyperoval, henceqeven, and characterize codewords of small weight. When investigating the small weight codewords of$T_2^*(\Theta)^D$, we deal with the case of$\Theta$a regular hyperoval, that is, a conic and its nucleus, separately, since in this case, we have a larger upper bound on the weight for which the results are valid. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 1–24, 2009