Recent Developments in Low-Density Parity-Check Codes

In this paper we prove two results related to low-density parity-check (LDPC) codes. The first is to show that the generating function attached to the pseudo-codewords of an LDPC code is a rational function, answering a question raised in [6]. The combinatorial information of its numerator and denominator is also discussed. The second concerns an infinite family of q-regular bipartite graphs with large girth constructed in [8]. The LDPC codes based on these graphs have attracted much attention. We show that the first few of these graphs are Ramanujan graphs.