Construction of LDPC Codes from Ramanujan Graphs

The use of iterative decoding algorithms for low-density parity check (LDPC) codes on bipartite graphs, and in general for codes on graphs, is capable of achieving near-Shannon limit performance. Unlike non-iterative decoding algorithms where the minimum distance of the code is a major factor in determining the code performance, it is the match of the graph extremal properties such as girth and expanding coefficient with the needs of iterative decoding that determines the code performance. We propose an algebraic construction method of asymptotically good (ir)regular LDPC codes matched to the needs of the iterative decoder based on Ramanujan graphs. Unlike other similar methods available in the literature, our technique allows for constructing a wider range of LDPC codes of length < m by eliminating the constraint on m being a prime. This is achieved by defining the Ramanujan graphs as Cayley graphs of an appropriately chosen subset of the group of 2×2 invertible matrices PGL2(m) over the ring Zm. The proposed method reduces to the currently known method when this subset becomes all of PGL2(m) when m is prime.