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IEE Proceedings - Communications

Acyclic Tanner graphs and maximum-likelihood decoding of linear block codes

Morteza Esmaeili, Amir Keyvan Khandani

Published Jan 1, 2000• 8 citations
Publisher

Abstract

The maximum-likelihood decoding of linear block codes by Wagner rule decoding is discussed. In this approach, the Wagner rule decoding, which has been primarily applied to single parity check codes, is employed on acyclic Tanner graphs. Accordingly, a coset decoding equipped with Wagner rule decoding is applied to the decoding of a code C having a Tanner graph with cycles. A subcode C1 of C with acyclic Tanner graph is chosen as the base subcode. All cosets of C1 have the same Tanner graph and are distinguished by their values of parity nodes in the graph. The acyclic Tanner graph of C1, together with a trellis representation of the space of the parity sequences, represent the code C. This graphical representation provides a unified and systematic approach to search for an efficient method for the maximum-likelihood decoding of a given linear block code. It is shown that the proposed method covers the most efficient techniques known for the decoding of some important block codes, including the hexacode H6, extended Golay codes, Reed–Muller codes, Hamming code and (32, 16, 8) quadratic residue codes. The generalisation to the decoding of lattices is briefly explained.

DOI
10.1049/ip-com:20000663

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The maximum-likelihood decoding of linear block codes by Wagner rule decoding is discussed. In this approach, the Wagner rule decoding, which has been primarily applied to single parity check codes, is employed on acyclic Tanner graphs. Accordingly, a coset decoding equipped with Wagner rule decoding is applied to the decoding of a code C having a Tanner graph with cycles. A subcode C1 of C with acyclic Tanner graph is chosen as the base subcode. All cosets of C1 have the same Tanner graph and are distinguished by their values of parity nodes in the graph. The acyclic Tanner graph of C1, together with a trellis representation of the space of the parity sequences, represent the code C. This graphical representation provides a unified and systematic approach to search for an efficient method for the maximum-likelihood decoding of a given linear block code. It is shown that the proposed method covers the most efficient techniques known for the decoding of some important block codes, including the hexacode H6, extended Golay codes, Reed–Muller codes, Hamming code and (32, 16, 8) quadratic residue codes. The generalisation to the decoding of lattices is briefly explained.
We present a method for maximum likelihood decoding of the (48,24,12) quadratic residue code. This method is based on projecting the code onto a subcode with an acyclic Tanner graph, and representing the set of coset leaders by a trellis diagram. This results in a two level coset decoding which can be considered a systematic generalization of the Wagner rule. We show that unlike the (24,12,8) Golay code, the (48,24,12) code does not have a Pless-construction which has been an open question in the literature. It is determined that the highest minimum distance of a (48,24) binary code having a Pless (1986) construction is 10, and up to equivalence there are three such codes.
Product codes have been an effective coding method for communication channels where both random and burst error occur. We present a new approach to the structure and maximum likelihood (ML) decoding of product codes using Tanner (1981) graphs. For product codes having a sub-code which is a product of simple parity codes and repetition codes, we show how to obtain a sub-code with an acyclic Tanner graph and the largest possible distance. We show that in all cases of interest, a n-dimensional product code has such a structure. Wagner rule decoding is used on this sub-code and its cosets to obtain an effective and efficient maximum likelihood decoding of the given product code.
Maximum-likelihood soft-decision decoding of linear block codes is addressed. A binary multiple-check generalization of the Wagner rule is presented, and two methods for its implementation, one of which resembles the suboptimal Forney-Chase algorithms, are described. Besides efficient soft decoding of small codes, the generalized rule enables utilization of subspaces of a wide variety, thereby yielding maximum-likelihood decoders with substantially reduced computational complexity for some larger binary codes. More sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the (24, 12) Golay code, yielding a computational gain factor of about 2 with respect to previous methods. A ternary single-check version of the Wagner rule is applied for efficient soft decoding of the (12, 6) ternary Golay code.>
It is widely accepted that short cycles in Tanner graphs deteriorate the performance of message-passing algorithms. This discourages the use of these algorithms on Tanner graphs (TGs) of well-known algebraic codes such as Hamming codes, Bose-Chaudhuri-Hocquenghem codes, and Reed-Solomon codes. Yedidia et al. presented a method to generate code representations suitable for message-passing algorithms. This method does not guarantee a representation free of four-cycles. In this correspondence, we present an algorithm to convert an arbitrary linear block into a code with orthogonal parity-check equations. A combinatorial argument is used to prove that the algorithm guarantees a four-cycle free representation for any linear code. The effects of removing four-cycles on the performance of a belief propagation decoder for the binary erasure channel are studied in detail by analyzing the structures in different representations. Finally, we present bit-error rate (BER) and block-error rate (BLER) performance curves of linear block codes under belief propagation algorithms for the binary erasure channel and the additive white Gaussian noise (AWGN) channel in order to demonstrate the improvement in performance achieved with the help of the proposed algorithm.
In decoding a linear block code C by iterative decoding algorithms, these algorithms are essentially applied on a graphical representation of C such as Tanner graph and factor graph. Due to the impact of the structure of these graphs on the performance of code, we consider minimal parity check matrices(matrices with minimum number of nonzero entries) of product codes. An algorithm constructing such matrices is given. It turns out that under the given construction method, product coding technique is indeed a powerful method to construct irregular LDPC codes. Given two codes A and B, the construction method produces a Tanner graph with girth g := min {8, ga, gb} for the product code A ⊗ B, where ga and gb are the girth of Tanner graphs representing A and B, respectively. Simulation results confirm the positive practical impact of the minimality of the parity check matrices.