Iterative decoding algorithms based on the alternating direction method of multipliers (ADMM) decoding of low density parity check (LDPC) codes has emerged as an alternating decoding method and bringed a boom of research on drawing upon mathematical optimization to LDPC decoding. Improving error-correcting performance is a key issue to enhance the superiority of ADMM decoding. In this letter, we investigate an efficient ADMM-based iterative decoder for binary LDPC codes. First, we build an mathematical programming equivalence of the maximum likelihood (ML) decoding problem by transforming parity-check constraints to multiple equivalent linear constraints and eliminating check-polytope projection (CPP). Then, an iterative algorithm based on ADMM technique is developed to solve this free-of-CPP (FCPP) equivalence and each ADMM update can be computed efficiently. Moreover, the proposed ADMM-FCPP decoding algorithm is analyzed to display a linear complexity to the length of the LDPC code at each iteration. Finally, simulation results demonstrate the superiority of the proposed decoder in error-correcting performance compared with the state-of-the-art ADMM-based decoders.

In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-ADMM iteration scales linearly with LDPC code’s length and the size of the considered Galois field. Moreover, the decoder based on the Constant-Weight embedding technique satisfies the favorable property of codeword symmetry. Simulation results demonstrate their effectiveness in comparison with state-of-the-art LDPC decoders.

In this paper, we develop an efficient quadratic programming (QP) decoding algorithm via the alternating direction method of multipliers (ADMM) technique for binary low density parity check (LDPC) codes. Its main content is as follows: first, through transforming the three-variables parity check equation to its equivalent expression, we relax the maximum likelihood decoding problem to a quadratic program. Second, the ADMM technique is exploited to design the solving algorithm of the resulting QP decoding model. Compared with the existing ADMM-based mathematical programming (MP) decoding algorithms, our proposed algorithm eliminates complex Euclidean projection onto the check polytope. Third, we prove that the proposed algorithm satisfies the favorable property of all-zeros assumption. Moreover, by exploiting the inside structure of the QP model, we show that the decoding complexity of our proposed algorithm in each iteration is linear in terms of LDPC code length. Simulation results demonstrate that the proposed QP decoder attains better error-correction performance than the sum-product BP decoder and costs the least amount of decoding time amongst the state-of-the-art ADMM-based MP decoding algorithms.