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 . The combinatorial information of its numerator and denominator is also discussed.
The second concerns an infinite family of
-regular bipartite graphs with large girth constructed in . The LDPC codes based on these graphs have attracted much attention. We show that the first few of these graphs are Ramanujan graphs.