The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.

The complexity-rate tradeoff for error-correcting codes below the Shannon limit is a central question in coding theory. This paper makes progress in this area by presenting a joint numerical optimization of rate and decoding complexity for low-density parity-check codes. The focus of this paper is on the binary symmetric channel and on a class of decoding algorithms for which an exact extrinsic information transfer (EXIT) chart analysis is possible. This class of decoding algorithms includes the Gallager decoding algorithm B. The main feature of the optimization method is a complexity measure based on the EXIT chart that accurately estimates the number of iterations required for the decoding algorithm to reach a target error rate. Under a fixed check-degree distribution, it is shown that the proposed complexity measure is a convex function of the variable-degree distribution in a region of interest. This allows us to numerically characterize the complexity-rate tradeoff. We show that for the Gallager B decoding algorithm on binary symmetric channels, the optimization procedure can produce complexity savings of 30-40% as compared to the conventional code design method