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International Conference on Intelligent Computing and Signal Processing (ICSP)

ℓ2-box ADMM Decoder for Non-binary LDPC Codes in F2q

Xiaomeng Guo, Xiaohe Li, Baoyi Xu, Jiangtao Wang

Published May 16, 2025• 0 citations
Publisher

Abstract

In this paper, we devise a novel decoding approach for Galois fields (${\mathbb{F}_{{2^q}}}$)-based non-binary Low-Density Parity-Check (LDPC) codes, leveraging the Alternating Direction Method of Multipliers (ADMM). The development of this decoder is structured around three primary contributions: firstly, we reformulate the Maximum Likelihood (ML) decoding problem as a discrete optimization task in Euclidean space, facilitated by the bit-embedding technique; secondly, we demonstrate the conversion of this integer problem into a continuous format via the ℓp-box method, for which a customized ADMM algorithm is developed; thirdly, we theoretical verify the proposed decoder’s codeword-independent property with per-iteration operations scaling approximately linearly with the code dimension (denoted as $\mathcal{O}(nq)$). Simulation results show that the error-correction capability and decoding efficiency is very competitive in comparison with contemporary non-binary LDPC decoders.

DOI
10.1109/icsp65755.2025.11087180

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In this paper, we devise a novel decoding approach for Galois fields (F2q{\mathbb{F}_{{2^q}}}F2q​)-based non-binary Low-Density Parity-Check (LDPC) codes, leveraging the Alternating Direction Method of Multipliers (ADMM). The development of this decoder is structured around three primary contributions: firstly, we reformulate the Maximum Likelihood (ML) decoding problem as a discrete optimization task in Euclidean space, facilitated by the bit-embedding technique; secondly, we demonstrate the conversion of this integer problem into a continuous format via the ℓp-box method, for which a customized ADMM algorithm is developed; thirdly, we theoretical verify the proposed decoder’s codeword-independent property with per-iteration operations scaling approximately linearly with the code dimension (denoted as O(nq)\mathcal{O}(nq)O(nq)). Simulation results show that the error-correction capability and decoding efficiency is very competitive in comparison with contemporary non-binary LDPC decoders.
In this paper, we devise a new alternating direction method of multipliers (ADMM) decoder for non-binary low-density parity-check (LDPC) codes in Galois fields of characteristic two (F2q)\left( {{\mathbb{F}_{{2^q}}}} \right)(F2q​). Its main content are threefold: first, the procedure of formulating the maximum likelihood (ML) decoding problem in F2q{\mathbb{F}_{{2^q}}}F2q​ to a linear integer problem in real space is presented; Second, after relaxing the integer problem to a continuous one, an efficient ADMM algorithm is customized to solve the latter, where all the entries of the variable vectors can be obtained in parallel; Third, we show that the proposed ADMM decoder satisfies the favorable codeword-independent property under some mild conditions and its computation complexity in each ADMM iteration is roughly O(nq)\mathcal{O}\left( {nq} \right)O(nq), where n is code length of the considered non-binary LDPC code. Simulation results demonstrate that its performance, such as error-correction and decoding efficiency, is very competitive in comparison with state-of-the-art non-binary LDPC decoders.
In this paper, we devote to devise a non-binary low-density parity-check (LDPC) decoder in Galois fields of characteristic two (F2q\mathbb{F}_{2^q}F2q​) via the alternating direction method of multipliers (ADMM) technique. Through the proposed bit embedding technique and the decomposition technique of the three-variables parity-check equation, an efficient ADMM decoding algorithm for non-binary LDPC codes is proposed. The computation complexity in each ADMM iteration is roughly O(nq)\mathcal{O}(nq)O(nq), which is significantly lower than the existing LDPC decoders. Moreover, we prove that the proposed decoder satisfies the favorable property of the codeword-independent. Simulation results demonstrate the outstanding performance of the proposed decoder in contrast with state-of-the-art LDPC decoders.
ADMM LP decoding of non-binary LDPC codes in F2m\mathbb{F}_{2^m}F2m​
In this paper, we develop efficient decoders for non-binary low-density parity-check (LDPC) codes using the alternating direction method of multipliers (ADMM). We apply ADMM to two decoding problems. The first problem is linear programming (LP) decoding. In order to develop an efficient algorithm, we focus on non-binary codes in fields of characteristic two. This allows us to transform each constraint in F2m\mathbb{F}_{2^m}F2m​ to a set of constraints in F2\mathbb{F}_{2}F2​ that has a factor graph representation. Applying ADMM to the LP decoding problem results in two types of non-trivial sub-routines. The first type requires us to solve an unconstrained quadratic program. We solve this problem efficiently by leveraging new results obtained from studying the above factor graphs. The second type requires Euclidean projection onto polytopes that are studied in the literature, a projection that can be solved efficiently using off-the-shelf techniques, which scale linearly in the dimension of the vector to project. ADMM LP decoding scales linearly with block length, linearly with check degree, and quadratically with field size. The second problem we consider is a penalized LP decoding problem. This problem is obtained by incorporating a penalty term into the LP decoding objective. The purpose of the penalty term is to make non-integer solutions (pseudocodewords) more expensive and hence to improve decoding performance. The ADMM algorithm for the penalized LP problem requires Euclidean projection onto a polytope formed by embedding the constraints specified by the non-binary single parity-check code, which can be solved by applying the ADMM technique to the resulting quadratic program. Empirically, this decoder achieves a much reduced error rate than LP decoding at low signal-to-noise ratios.
In this paper, we develop efficient decoders for non-binary low-density parity-check codes using the alternating direction method of multipliers (ADMM). We apply ADMM to two decoding problems. The first problem is linear programming (LP) decoding. In order to develop an efficient algorithm, we focus on non-binary codes in fields of characteristic two. This allows us to transform each constraint in F2m to a set of constraints in F2that has a factor graph representation. Applying ADMM to the LP decoding problem results in two types of non-trivial sub-routines. The first type requires us to solve an unconstrained quadratic program. We solve this problem efficiently by leveraging new results obtained from studying the above factor graphs. The second type requires Euclidean projections onto polytopes that are studied in the literature. Such projections can be solved efficiently using off-the-shelf techniques, which scale linearly in the dimension of the vector to project. ADMM LP decoding scales linearly with block length, linearly with check degree, and quadratically with field size. The second problem we consider is a penalized LP decoding problem. This problem is obtained by incorporating a penalty term into the LP decoding objective. The purpose of the penalty term is to make non-integer solutions (pseudocodewords) more expensive and hence to improve decoding performance. The ADMM algorithm for the penalized LP problem requires Euclidean projection onto a polytope formed by embedding the constraints specified by the non-binary single parity-check code, which can be solved by applying the ADMM technique to the resulting quadratic program. Empirically, this decoder achieves a much reduced error rate than LP decoding at low signal-to-noise ratios.
In this paper, we develop an efficient decoder via the proximal alternating direction method of multipliers (proximal-ADMM) technique for nonbinary linear block codes in the Galois field. Its main contents are as follows: first, exploiting the decomposition technique based on the three-variables check equation, we formulate the maximum likelihood (ML) decoding problem approximately to a non-convex quadratic program; second, an efficient algorithm based on the proximal-ADMM technique is proposed to solve the formulated QP problem. Exploiting the QP problem's inherent structures, its variables can be updated in parallel; third, we prove that the proposed decoding algorithm can converge to some stationary point of the formulated QP problem. Moreover, we also show, for nonbinary low-density parity-check (LDPC) codes, its computational complexity in each proximal-ADMM iteration scales linearly with block length and the size of the considered Galois field. Simulation results demonstrate that the proposed proximal-ADMM decoder outperforms state-of-the-art nonbinary LDPC decoders in terms of either error correction performance or computational complexity.
In this paper, we develop an efficient algorithm for linear programming (LP) decoding of non-binary low-density parity-check (LDPC) codes. We build our algorithm on the decomposition method based on the alternating direction method of multipliers (ADMM). Although expressing the LP decoding problem using ADMM is not hard, a sub-routine of ADMM - projection onto a polytope formed from embeddings of the non-binary single parity-check code - is not straightforward. In this work, we focus on non-binary codes in fields of characteristic two. This allows us to use operations in F2to relax the polytope under consideration into a form that is computational friendly. We introduce a rotation step that normalizes the geometry under consideration. We then apply ADMM a second time to solve the projection problem, which is a quadratic program (QP). Our decoding algorithm scales linearly with block length, linearly with check degree, and quadratically with field size.