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IET Communications

Baseline parity‐check matrix for iterative soft‐decision decoding of binary cyclic codes

Oluwaseyi Paul Babalola, Olayinka Olaolu Ogundile, D.J.J. Versfeld

Published Apr 17, 2020• 1 citation
Publisher

Abstract

Belief propagation (BP) is a soft‐decision (SD) decoder that is commonly used to obtain near‐optimal decoding performance for linear codes defined over sparse parity‐check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity‐check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity‐check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity‐check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub‐optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity‐check equations instead of maximum‐likelihood decoding.

DOI
10.1049/iet-com.2019.1008

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Belief propagation (BP) is a soft‐decision (SD) decoder that is commonly used to obtain near‐optimal decoding performance for linear codes defined over sparse parity‐check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity‐check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity‐check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity‐check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub‐optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity‐check equations instead of maximum‐likelihood decoding.
In this paper, an iterative soft-decision (SD) decoding algorithm for cyclic codes based on extended parity-check equations is developed. The algorithm does not necessarily utilize the algebraic properties of the code, but operates on transforming the systematic parity-check matrix using the soft reliability information matrix obtained from the received vector. Results show a significant performance gain when compared with the hard decision Berlekamp-Massey(B-M) and belief propagation (BP) algorithms, but present a similar bit error rate (BER) performance when compared to the adaptive belief propagation (ABP) algorithm. An important feature of the decoder is that it functions within a practical decoding time complexity, and can be generally implemented for the class of linear block codes.
The conventional parity-check matrix transformation algorithm (PTA) requires matrix inversion to transform matrices of maximum distance separable (MDS) codes. However, such matrix transformation is not always guaranteed for the class of non-MDS codes. Hence, the PTA fails for binary cyclic (BC) codes. To overcome this limitation, we developed a generalized parity-check matrix transformation (GPT) algorithm for binary cyclic codes. The GPT avoids the matrix inversion step of the PTA. It permutes the columns of the parity-check matrix based on the reliability information from the channel. Results show a significant bit error rate (BER) performance gain as compared to the existing PTA. It also presents a reasonable BER performance as compared to the other soft-decision (SD) decoding algorithms. In addition, the decoder functions within a practical decoding time complexity, particularly at the message passing stage.
The Tanner graph associated with an extended parity-check (EPC) matrix of a cyclic code is shown to be useful in effectively implementing soft decision iterativedecoding procedures based on belief propagation. Decoding with an EPC matrix has the advantage that is universal, in the sense that it does not depend on the specific family of cyclic codes being used. It is shown that there is no need to store the complete EPC matrix, i.e., the structure of the Tanner graph over which iterative decoding is implemented. The length, dimension and parity-check polynomial are all that is needed as input parameters to the decoder. Iterative soft decision decoding can be implemented with a pair of processing elements, to pass messages between nodes in the graph, with edges specified by the parity-check polynomial. By identifying received word positions with high reliability, using a designed threshold, decoding complexity can be reduced drastically while maintaining good error performance. Keywords --- Iterative decoding, cyclic codes, Tanner graphs. 1 1
We introduce a new method for decoding short and moderate-length linear block codes with dense parity check matrix representations of cyclic form. This approach is termed multiple-bases belief-propagation. The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the multiple-bases belief propagation decoding performance can be shown to closely follow that of the maximum-likelihood decoder.
Belief propagation (BP) decoding has been recognized for its capacity-approaching performance and high throughput when decoding long low-density parity-check (LDPC) codes. However, the application of BP decoding for short codes is hindered by dense parity-check matrices (PCMs) and prevalent short cycles in the Tanner graph. In this paper, we introduce a general method to extract an optimized sparse PCM for short binary block codes, which removes length-four cycles and enhances the connectivity of short cycles to enable BP decoding with improved performance. Notably, for short binary codes with lengths up to 64, our BP decoding performance approaches the maximum likelihood bound and surpasses the best-reported BP results with reduced computational complexity. Compared with other universal decoding algorithms, BP decoding using our extracted sparse PCMs is competitive in terms of both error-rate performance and computational complexity. These promising results suggest that our method to improve BP decoding for short codes is a step toward a practical universal BP decoder for next-generation communication systems.
In this paper, we present a novel approach to improve belief propagation (BP) decoding performance by sparsifying the Tanner graph. Our proposed method first constructs a low-weight parity-check (LWPC) matrix with increased rows based on the original parity-check matrix. Then, all 4-cycles in the LWPC matrix are eliminated by adding auxiliary variable nodes and corresponding parity-check rows, resulting in the generalized LWPC (G-LWPC) matrix. Compared to the original matrix, the sparsity is greatly improved without altering the encoding constraints. Simulation results show that BP decoding based on the G-LWPC matrix is particularly effective for short code lengths. For a (128,20) polar code with 24-bit cyclic redundancy check, our proposed G-LWPC matrix reduces the density from 17.4% to 1.3%, and with 10 iterations, BP decoding can achieve the performance of successive cancellation list decoding with a list size of 32.