Algebraic Decoding of the Quadratic Residue Code
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 inversefree BerlekampMassey (BM) algorithm to determine the errorlocator 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 errorcorrecting 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.

 "Using a technique similar to that given in [11], a strategy in [17] was developed to obtain each of the needed primary unknown syndromes. It is based on solving the roots of the equation in a primary unknown syndrome with the coefficients that can be expressed in terms of certain primary known syndromes. "
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ABSTRACT: In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified ChaseII algorithm, called a new softdecision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the errorcorrecting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LPbased decoding improves the errorrate performance almost without increasing the decoding complexity, when compared with the new softdecision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 106.IEEE Transactions on Communications 03/2013; 61(3):832841. DOI:10.1109/TCOMM.2012.122712.120287 · 1.99 Impact Factor 
 "Using a technique similar to that given in [11], a strategy in [17] was developed to obtain each of the needed primary unknown syndromes. It is based on solving the roots of the equation in a primary unknown syndrome with the coefficients that can be expressed in terms of certain primary known syndromes. "
IEEE Transactions on Communications 01/2013; · 1.99 Impact Factor 
 ", e(β rs ))  e(x) ∈ E}. When C is the binary quadratic residue code of length 89, R C = {1, 5, 9, 11}, it has been proved [6] "
Article: Multivariate Interpolation Formula over Finite Fields and Its Applications in Coding Theory
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ABSTRACT: A multivariate interpolation formula (MVIF) over finite fields is presented by using the proposed Kronecker delta function. The MVIF can be applied to yield polynomial relations over the base field among homogeneous symmetric rational functions. Besides the property that all the coefficients are coming from the base field, there is also a significant one on the degrees of the obtained polynomial; namely, the degree of each term satisfies certain condition. Next, for any cyclic codes the unknown syndrome representation can also be provided by the proposed MVIF and also has the same properties. By applying the unknown syndrome representation and the BerlekampMassey algorithm, onestep decoding algorithms can be developed to determine the error locator polynomials for arbitrary cyclic codes.