Codes and Iterative Decoding on Algebraic Expander Graphs

The notion of graph expansion was introduced as a tool in coding theory by Sipser and Spielman, who used it to bound the minimum distance of a class of low-density codes, as well as the performance of various iterative decoding algorithms for these codes. In spite of its usefulness in establishing theoretical bounds on iterative decoding, graph expansion has not been widely used to design codes. Instead, random graphs are the primary means used to obtain graphs for codes, raising the question of whether comparable performance can be achieved using explicit constructions. In this paper we investigate the use of explicit algebraic expander graphs and algebraic subcodes, and show that the resulting coding schemes achieve excellent performance, competitive with standard low-density paritycheck codes over a wide range of block lengths. Since the code constructions are based on graphs of groups, the Fourier transform can be used to obtain fast encoding algorithms for these codes.