Low Density Parity Check Codes over GF ( q ) Matthew

| Binary Low Density Parity Check (LDPC) codes have been shown to have near Shannon limit performance when decoded using a probabilis-tic decoding algorithm. The analogous codes deened over nite elds GF (q) of order q > 2 show signii-cantly improved performance. We present the results of Monte Carlo simulations of the decoding of inn-nite LDPC Codes which can be used to obtain good constructions for nite Codes. We also present empirical results for the Gaussian channel including a rate 1/4 code with bit error probability of 10 ?4 at E b =N0 = ?0:05dB. I. Introduction We consider a class of error correcting codes rst described by Gallager in 1962 2]. These recently rediscovered low density parity check (LDPC) codes are deened in terms of a sparse parity check matrix and are known to be asymptotically good for all channels with symmetric stationary ergodic noise 6]. Practical decoding of these codes is possible using an approximate belief propagation algorithm and near Shannon limit performance has been reported 7]. We consider the generalisation of binary LDPC codes to nite elds GF (q), q > 2, and demonstrate a signiicant improvement in empirical performance. Although little is known about the theoretical properties of the approximate belief propagation algorithm when applied to this decoding problem, Monte Carlo methods may be used to simulate the behaviour of the decoding algorithm applied to an innnite LDPC code. We have used such Monte Carlo results to design better codes for practical decoding. In section II we deene low density parity check codes and in section III we describe the decoding algorithm. Section IV presents the results of the Monte Carlo simulation and empirical decoding results are presented in section V.