Design of Irregular Graphs for Erasure Decoding∗

Low-density parity check codes (LDPC codes) were introduced by Gallager [2] and have received intensive study in the last few years, as advances in technology have made their implementation far more practical than when they were originally proposed. Basically, an LDPC code is a binary linear block code defined by a sparse m × n parity-check matrix H . Equivalently such a code can be thought of as defined by a bipartite graph, called a Tanner graph, whose adjacency matrix is H . In this representation the n left vertices represent columns of H , corresponding to bits in the original message, and the m right vertices represent the rows of H , corresponding to parity check bits; an edge (i, j) exists in the graph whenever the corresponding entry hi,j in H is 1, i.e. whenever the message bit i participates in parity check j. Decoding of LDPC codes is then performed via an iterative algorithm operating on the Tanner graph. See the original paper [11] or the more recent work in [7] for the background on LDPC codes and more general codes on Tanner graphs. In this paper we will consider a Tanner graph, parity check matrix and the LDPC code they define to be more or less interchangeable (despite the fact that any linear code can in fact have multiple Tanner graph presentations.) In the original work Gallager was working with bipartite graphs which were regular. This means there were positive integers λ, ρ such that every left vertex had degree λ and every right vertex had degree ρ. Such a code is referred to as (λ, ρ)-regular.