Low density parity check codes based on sparse matrices with no small cycles
In this paper we give a systematic construction of matrices with constant row weights and column weights and arbitrarily large girths. This resolves a problem raised by D. MacKay. The matrices are used in the generator matrices of linear codes. We give the experiment performance results for codes whose associated matrices have girth 8. We also give a randomized construction of matrices with constant row sums and column sums and few 4-cycles. The codes generated using the matrices are used to encode bit streams for a Gaussian channel and decoded using a decoding algorithm that combines features of the algorithms given by MacKay and Cheng and McEliece. The experimental performance results for codes generated using the random matrices are compared to those of the systematically constructed codes. The results show that the codes generated using the random codes with smaller block sizes perform as well as the systematic codes with bigger block sizes. The performance of the systematic codes, for specified weights, can be used to tailor the random codes. MATLAB routines for the construction for the girth 8 case and a special girth 4 case are included.
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