Binary weight distribution of non-binary LDPC codes

Iryna Andriyanova (1) , Vishwambhar Rathi (2) , Jean-Pierre Tillich (3)

1 ETIS - UMR 8051 - Equipes Traitement de l'Information et Systèmes
2 KTH School of Electrical Engineering
3 SECRET - Security, Cryptology and Transmissions

This paper is the first part of an investigation if the capacity of a binary-input memoryless symmetric channel under ML decoding can be achieved asymptotically by using non-binary LDPC codes. We consider (l, r)-regular LDPC codes both over finite fields and over the general linear group and compute their asymptotic binary weight distributions in the limit of large blocklength and of large alphabet size. A surprising fact, the average binary weight distributions that we obtain do not tend to the binomial one for values of normalized binary weights ¿ smaller than 1-2-l/r. However, it does not mean that non-binary codes do not achieve the capacity asymptotically, but rather that there exists some exponentially small fraction of codes in the ensemble, which contains an exponentially large number of codewords of poor weight. The justification of this fact is beyond the scope of this paper and will be given in.

Domaines: Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]

Iryna Andriyanova, Vishwambhar Rathi, Jean-Pierre Tillich. Binary weight distribution of non-binary LDPC codes. IEEE International Symposium on Information Theory : ISIT 2009, Jul 2009, Seoul, South Korea. pp.65 - 69, ⟨10.1109/ISIT.2009.5205662⟩. ⟨hal-00524696⟩