Algebraic Decoding of the Quadratic Residue Code

IEEE Transactions on Information Theory (Impact Factor: 2.62). 01/2008; 54(11):5005-5011. DOI: 10.1109/TIT.2008.929956
Source: DBLP

ABSTRACT Recently, an algebraic decoding algorithm suggested by Truong (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100 , is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations.

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    ABSTRACT: A high-speed and memory-efficient table lookup decoding algorithm (TLDA), called the syndrome and syndrome difference decoding algorithm (SSDDA), is developed to decode the long binary systematic (71, 36, 11) quadratic residue (QR) code. The essential point of the SSDDA is based on the property of the weight of syndrome and the weight of syndrome difference to reduce the memory size of the lookup table. The proposed algorithm generates a novel compact lookup table (CLT), which only consists of 7,806 syndromes and their corresponding error patterns. Consequently, the memory size of the proposed CLT is only about 38.39% of the lookup table proposed by Lin et al.
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    IEEE Transactions on Communications 01/2013; · 1.75 Impact Factor
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    ABSTRACT: In this paper, a fast algebraic decoding algorithm (ADA) is proposed to correct all patterns of five errors or less in the binary systematic (71, 36, 11) quadratic residue (QR) code. The method is based on the modification of the ADAs developed by Reed et al and Lin et al. The new conditions and the error-locator polynomials for decoding this code will be derived. Besides, a computer search shows that the minimum degree of the unknown syndrome polynomial f(S7) in the five-error case is 2. Hence, the computational complexity can be reduced in finite field. Simulation result shows that the average decoding time of the proposed ADA is superior to the ADA given by Chang et al.