Design of Low-Density Parity-Check Codes with Optimized Complexity-Rate Tradeoff

The optimal complexity-rate tradeoff for errorcorrecting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the joint optimization of rate and 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. Under certain mild conditions, the complexity measure is a convex function of the variable edge-degree distribution of the code, allowing an efficient design of complexity-optimized LDPC codes using convex optimization methods. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform thresholdoptimized codes at long block lengths.