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IEEE Transactions on Information Theory

On the pless-construction and ML decoding of the (48, 24, 12) quadratic residue code

Morteza Esmaeili, Thomas Aaron Gulliver, Amir Keyvan Khandani

Published May 29, 2003• 4 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2003.811930

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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.
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.
Let E/sub m/ denote the binary (m, m-1, 2) even weight code and /spl ominus/ the Kronecker product operation. A linear code of length m/spl times/t containing E/sub m//spl ominus/(t,1,t) can be decoded efficiently using a Tanner graph and a trellis diagram together with the Wagner decoding algorithm. Thus, it is important to determine whether a given code contains such a subcode. We show that the (18,9,6) binary quadratic residue code does not contain E/sub 6//spl ominus/(3,1,3). The maximum minimum distance of such a code is 5.
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.
The Pless construction of the (24, 12, 8) extended Golay code can be modified to construct the (32, 16, 8) extended QR code. Some structural similarities are discovered between the (32, 16, 8) QR code and the (24, 12, 8) Golay code. A simple decoder is developed based on the two-level structure of the (32, 16, 8) QR code. In addition to correcting all three or less bit errors, the proposed algorithm is able to correct some four-bit error patterns. Recently, soft-logic decoding of block codes has found increasing practical applications. The two-level structure renders an extension of Vardy and Be‘ery’s fast algorithm to the (32, 16, 8) QR code. The Vardy-Be‘ery’s algorithm is, so far, the most efficient scheme for the Golay code in the literature, with direct application impact. Its extension to the (32, 16, 8) code is also computationally efficient, as analysed in detail.
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.>