Probabilistic Decoding of Low-Density Cayley Codes

We report on some investigations into the behavior of a class of low-density codes constructed using algebraic techniques. Recent work shows expansion to be an essential property of the graphs underlying the low-density parity-check codes first introduced by Gallager. In addition, it has recently been shown that certain spectral techniques similar to those based on Fourier analysis for classical cyclic codes can be applied to codes constructed from Cayley graphs. This motivates us to compare the behavior of algebraically constructed expanders and randomly generated bipartite graphs using a probabilistic decoding algorithm. Preliminary results indicate that the performance of the explicit, algebraic expanders is comparable to that of random graphs in the case where each variable is associated with only two parity checks, while such codes are inferior to randomly generated codes with three or more constraints for each variable.