Average Spectra and Minimum Distances of Low-Density Parity-Check Codes over Abelian Groups
SIAM Journal on Discrete Mathematics
Performance bounds for nonbinary linear block codes over memoryless symmetric channels
IEEE Transactions on Information Theory
Design of capacity-approaching irregular low-density parity-check codes
IEEE Transactions on Information Theory
On the application of LDPC codes to arbitrary discrete-memoryless channels
IEEE Transactions on Information Theory
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This paper is the first part of an investigation if the capacity of a binary-input memoryless symmetric channel under ML decoding can be achieved asymptotically by using non-binary LDPC codes. We consider (l, r)-regular LDPC codes both over finite fields and over the general linear group and compute their asymptotic binary weight distributions in the limit of large blocklength and of large alphabet size. A surprising fact, the average binary weight distributions that we obtain do not tend to the binomial one for values of normalized binary weights ω smaller than 1-2-l/r. However, it does not mean that non-binary codes do not achieve the capacity asymptotically, but rather that there exists some exponentially small fraction of codes in the ensemble, which contains an exponentially large number of codewords of poor weight. The justification of this fact is beyond the scope of this paper and will be given in [1].