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IEEE International Symposium on Information Theory (ISIT)

ADMM decoding of non-binary LDPC codes in F2m

Xishuo Liu, Stark Christiaan Draper

Published Jun 1, 2014• 2 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2014.6875274

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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.
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.
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.
This paper presents an efficient quadratic programming (QP) decoder via the alternating direction method of multipliers (ADMM) technique, called QP-ADMM, for binary low-density parity-check (LDPC) codes. Its main contents are as follows: first, we relax the maximum likelihood (ML) decoding problem to a non-convex quadratic program. Then, we develop an ADMM solving algorithm for the formulated non-convex QP decoding model. In the proposed QP-ADMM decoder, complex Euclidean projections onto the check polytope are eliminated and variables in each updated step can be solved analytically in parallel. Moreover, it is proved that the proposed ADMM algorithm converges to a stationary point of the non-convex QP problem under the assumption of sequence convergence. We also verify that the proposed decoder satisfies the favorable property of the all-zeros assumption. Furthermore, by exploiting the inside structures of the QP model, the complexity of the proposed algorithm in each iteration is shown to be linear in terms of LDPC code length. Simulation results demonstrate the effectiveness of the proposed QP-ADMM decoder.
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.
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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.