B i n a r y Low D e n s i t y P a r i t y Check ( L D P C ) codes have been s h o w n t o h a v e near S h a n n o n limit performance w h e n d e c o d e d using a probabilistic decoding algorithm. T h e analogous codes defined over finite fields GF(q) of o r d e r q > 2 show significantly improved performance. We present the results of M o n t e Carlo simulations of the decoding of infin i t e L D P C Codes which c a n be used to o b t a i n g o o d constructions for finite Codes. We also present empirical results for the Gaussian channel including a r a t e 114 c o d e w i t h bit e r r o r probability of at EbINo = -0.05dB.

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The low density parity check codes whose performance is closest to the Shannon limit are`Gallager codes' based on irregular graphs. We compare alternative methods for constructing these graphs and present two results. First, we nd a`super{Poisson' construction which gives a small improvement in empirical performance over a random construction. Second, whereas Gallager codes normally take N 2 time to encode, we investigate constructions of regular and irregular Gallager codes which allow more rapid encoding and have smaller memory requirements in the encoder. We nd that thesè fast{encoding' Gallager codes have equally good performance.