Low Density Parity Check Codes

Abstract

A linear code with a check matrix in which each column has few non-zero entries is called a low density parity check code or, for brevity, an LDPC code. These codes were introduced in the 1960s by Gallager who proved that probabilistic constructions of such matrices produce asymptotically good linear codes. Moreover, he observed that LDPC codes perform well when applying the following decoding algorithm. On receiving a vector v, one calculates the weight of the syndrome of v + e, for each vector e of weight one. If the weight of this syndrome is less than the weight of the syndrome of v, for some e, then we replace v by v + e and repeat the process. If at each iteration there is such a vector e, then, since after replacing v by v + e, the weight of the syndrome of v decreases, we will eventually find a vector whose syndrome is zero, which must be the syndrome of some codeword u. We then decode v as u. If at some iteration no such e exists then the decoding breaks down. If at some iteration more than one such vector e exists, then one could choose e so that the weight of the syndrome of v + e is minimised. In this chapter we will prove that there are LDPC codes, constructed from graphs with the expander property, for which the decoding algorithm will not break down. Provided that the number of error bits is less than half the minimum distance, the decoding algorithm will return the nearest codeword to the received vector. We will use probabilistic arguments to construct the graphs, and from these a sequence of codes which are asymptotically good.