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arXiv:1403.7851 · IEEE International Symposium on Information Theory (ISIT)

Adaptive linear programming decoding of polar codes

Veeresh Taranalli, Paul H. Siegel

Published Jun 1, 2014• 7 citations
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Abstract

Polar codes are high density parity check codes and hence the sparse factor graph, instead of the parity check matrix, has been used to practically represent an LP polytope for LP decoding. Although LP decoding on this polytope has the ML-certificate property, it performs poorly over a BAWGN channel. In this paper, we propose modifications to adaptive cut generation based LP decoding techniques and apply the modified-adaptive LP decoder to short block-length polar codes over a BAWGN channel. The proposed decoder provides significant FER performance gain compared to the previously proposed LP decoder and its performance approaches that of ML decoding at high SNRs. We also present an algorithm to obtain a smaller factor graph from the original sparse factor graph of a polar code. This reduced factor graph preserves the small check node degrees needed to represent the LP polytope in practice. We show that the fundamental polytope of the reduced factor graph can be obtained from the projection of the polytope represented by the original sparse factor graph and the frozen bit information. Thus, the LP decoding time complexity is decreased without changing the FER performance by using the reduced factor graph representation.

DOI
10.1109/isit.2014.6875381

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Polar codes are high density parity check codes and hence the sparse factor graph, instead of the parity check matrix, has been used to practically represent an LP polytope for LP decoding. Although LP decoding on this polytope has the ML-certificate property, it performs poorly over a BAWGN channel. In this paper, we propose modifications to adaptive cut generation based LP decoding techniques and apply the modified-adaptive LP decoder to short block-length polar codes over a BAWGN channel. The proposed decoder provides significant FER performance gain compared to the previously proposed LP decoder and its performance approaches that of ML decoding at high SNRs. We also present an algorithm to obtain a smaller factor graph from the original sparse factor graph of a polar code. This reduced factor graph preserves the small check node degrees needed to represent the LP polytope in practice. We show that the fundamental polytope of the reduced factor graph can be obtained from the projection of the polytope represented by the original sparse factor graph and the frozen bit information. Thus, the LP decoding time complexity is decreased without changing the FER performance by using the reduced factor graph representation.
Polar codes are the first codes to provably achieve capacity on the symmetric binary-input discrete memoryless channel (B-DMC) with low encoding and decoding complexity. The parity check matrix of polar codes is high-density and we show that linear program (LP) decoding fails on the fundamental polytope of the parity check matrix. The recursive structure of the code permits a sparse factor graph representation. We define a new polytope based on the fundamental polytope of the sparse graph representation. This new polytope P is defined in a space of dimension O(N logN) where N is the block length. We prove that the projection of P in the original space is tighter than the fundamental polytope based on the parity check matrix. The LP decoder over P obtains the ML-certificate property. In the case of the binary erasure channel (BEC), the new LP decoder is equivalent to the belief propagation (BP) decoder operating on the sparse factor graph representation, and hence achieves capacity. Simulation results of SC (successive cancellation) decoding, LP decoding over tightened polytopes, and (ML) maximum likelihood decoding are provided. For channels other than the BEC, we discuss why LP decoding over P with a linear objective function is insufficient.
Linear programming (LP) decoding approximates optimal maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of linear inequalities derived from the constraints represented by the rows of a parity-check matrix of the code. Adaptive linear programming (ALP) decoding significantly reduces the complexity of LP decoding by iteratively and adaptively adding necessary constraints in a sequence of smaller LP problems. Adaptive introduction of constraints derived from certain additional redundant parity check (RPC) constraints can further improve ALP performance. In this paper, we propose a new and effective algorithm to identify RPCs that produce linear constraints, referred to as “cuts,” that can eliminate non-ML solutions generated by the ALP decoder, often significantly improving the decoder error-rate performance. The cut-finding algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. Simulation results for several low-density parity-check codes demonstrate that the modified ALP decoding algorithm significantly narrows the performance gap between LP decoding and ML decoding.
Abstract—Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities ” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut, ” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding. Index Terms—Iterative decoding, linear codes, linear programming (LP) decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, pseudocodewords. I.
Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut,” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding.
The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LOpe codes suggest that the M LP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.