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

Abstract

In this paper, 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 (Θ) and T * 2 (Θ) D with Θ a hyperoval, hence q even, and characterize codewords of small weight. When investigating the small weight codewords of T * 2 (Θ) D , we deal with the case of Θ a regular hyperoval, i.e. 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.

Some low-density parity-check codes derived from finite geometries

“…It has been proven in [21] that the plane words are exactly the code words of minimum weight, up to a scalar factor.…”

“…Under the same condition char K = 2, the same result will be established for this nonregular generalized quadrangle and its dimension will be computed. In [21], the authors manage to classify the code words of small weight for sufficiently large q, as a linear combination of code words of minimum weight but in some cases also second-minimum weight. For char K = 2 we show that all words in the code are a linear combination of the code words of minimum weight, even the code words of second minimum weight used in the classification of [21], regardless of weight and even for small q.…”

A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2(Q), Edoukou1, Hallez2, Rodier3 et al. 2010Journal of Pure and Applied AlgebraSelf Cite9000View full textAdd to dashboardCite

“…Some of these results were motivated by the recent interest in LDPC (Low Density Parity-check Codes) codes [2,11,[16][17][18]. In all of these cases, the problem of investigating the small weight codewords of these linear codes was translated into a geometrical problem in finite projective spaces, thereby illustrating the use of geometrical methods for obtaining new results on the small weight codewords in linear codes related to geometrical structures.…”

“…We wish however to mention that the study of small weight codewords in different classes of linear codes related to geometrical structures has received great attention in the recent literature. We mention in particular the results of [2] on linear codes defined by conics in PG (2, q), the results of [11,[16][17][18] on small weight codewords related to generalized quadrangles and classical polar spaces, the results on the small weight codewords of the linear codes related to the incidence matrices of PG(N, q) [7,[13][14][15]22], and the results on the small weight codewords of the d-th-order projective Reed-Muller codes PRM(q, d, n) [19][20][21].…”

Linear representations of finite geometries and associated LDPC codes

“…Each row of N has weight |K| and each column has weight q so that C and D are LDPC codes of lengths |K|q n−1 and q n , respectively, with dimensions given by Theorem 1.1. The code C has been studied in [13], [9] and [12]. In [14], Kou et al considered the codes C and D for F = F 2 and when K is the set of all points of H, referring to C as the type-II geometry-G LDPC code and D as the type-I geometry-G LDPC code.…”