Algebraic Decoding of the Quadratic Residue Code

IEEE Transactions on Information Theory (Impact Factor: 2.65). 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 general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several specific results for classes of cyclic codes. From these, a theoretically justification of the sparsity of the general error locator polynomial is obtained for all cyclic codes with $t\leq 3$ and $n<63$, except for three cases.
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    ABSTRACT: In this paper1, a method to search the subsets I and J needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting I and J, one computes the unknown syndromes, and thus finds the corresponding error-locator polynomial by using an inverse-free BM algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is never seen in the literature, to our knowledge. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity.
    ICC 2014 - 2014 IEEE International Conference on Communications; 06/2014
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    ABSTRACT: In this paper, the performance of quadratic residue (QR) codes of lengths within 100 is given and analyzed when the hard decoding, soft decoding, and linear programming decoding algorithms are utilized. We develop a simple method to estimate the soft decoding performance, which avoids extensive simulations. Also, a simulation-based algorithm is proposed to obtain the maximum likelihood decoding performance of QR codes of lengths within 100. Moreover, four important theorems are proposed to predict the performance of the hard decoding and the maximum-likelihood decoding in which they can explore some internal properties of QR codes. It is shown that such four theorems can be applied to the QR codes with lengths less than 100 for predicting the decoding performance. In contrast, they can be straightforwardly generalized to longer QR codes. The result is never seen in the literature, to our knowledge. Simulation results show that the estimated hard decoding performance is very accurate in the whole signal-to-noise ratio (SNR) regimes, whereas the derived upper bounds of the maximum likelihood decoding are only tight for moderate to high SNR regions. For each of the considered QR codes, the soft decoding is approximately 1.5 dB better than the hard decoding. By using powerful redundant parity-check cuts, the linear programming-based decoding algorithm, i.e., the ACG-ALP decoding algorithm performs very well for any QR code. Sometimes, it is even superior to the Chase-based soft decoding algorithm significantly, and hence is only a few tenths of dB away from the maximum likelihood decoding.