Abstract

This paper introduces methods based on generalized Fourier analysis for working with a class of errorcorrecting codes constructed in terms of Cayley graphs. Our work is motivated by the recent results of Sipser and Spielman [15] showing graph expansion to be essential for efficient decoding of certain low-density parity-check codes. They leave open the problem of sub-quadratic encoding for this class of codes, and it is this problem that we address. We show that when the codes are constructed in terms of Cayley graphs, the symmetry of the graphs can be exploited by using the representation theory of the underlying group to devise a sub-quadratic encoding algorithm that, in the case where the group is PSL 2 (Z=qZ), requires O(n 4=3 ) operations, where n = O(q 3 ) is the block length. Our results indicate that this new class of codes may combine many of the strengths of two of the most powerful and successful, but previously disparate areas of coding theory: the class of cyclic codes...

Citations