Quadratic Residue

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Trieu-kien Truong - One of the best experts on this subject based on the ideXlab platform.

  • Algebraic Decoding of Some Quadratic Residue Codes With Weak Locators
    IEEE Transactions on Information Theory, 2015
    Co-Authors: Yan-haw Chen, Trieu-kien Truong, Yaotsu Chang
    Abstract:

    In this paper, an explicit expression of the weak-locator polynomial for p-ary Quadratic Residue codes is presented by a modification of the Feng-Tzeng matrix method. The differences between the modified version and the original Feng-Tzeng matrix are that in the new matrix, not every entry is a syndrome, and every syndrome entry is a known syndrome. By utilizing this technique, an algebraic decoding of the ternary (61, 30, 12) Quadratic Residue code is proposed. This new result has never been seen in the literature to our knowledge. An advantage of the proposed decoding algorithm is that in general the obtained weak-locator polynomials can decode efficiently not only all the error patterns of weights four and five, but also some error patterns of weight six.

  • Comments on “On Decoding of the (89, 45, 17) Quadratic Residue Code”
    IEEE Transactions on Communications, 2015
    Co-Authors: Yong Li, Pengwei Zhang, Lin Wang, Trieu-kien Truong
    Abstract:

    Presents comments on the paper, "On decoding of the Quadratic Residue code,” (Wang, L., et al) IEEE Trans. Commun., vol. 61, no. 3, pp. 832–841, Mar. 2013.

  • On the Low-Complexity Soft-Decoding of Binary Quadratic Residue Codes
    Proceedings of the 1st IEEE IIAE International Conference on Intelligent Systems and Image Processing 2013, 2013
    Co-Authors: Wen-ku Su, Pei-yu Shih, Trieu-kien Truong
    Abstract:

    This paper is to shown the performance of a modified algorithm for decoding the (4 7 , 24, 1 1 ) binary Quadratic Residue code up to six errors. The technique in this paper combines the algebraic decoding algorithm with the detection scheme for the (47, 24, 11) Quadratic Residue code offered by Truong et al. to correct up to five errors. Then again, the reliability-search algorithm is utilized to correct one bit of the six bits error. The computer simulation of the scheme shows that at least 9 0 % of 6 bits error occurred are corrected if the E b /N 0 ratio s are greater than 4 dB. As a result, the performance of this modified algorithm is very close to the bound of decoding up to six-bit error.

  • decoding of the 31 16 7 Quadratic Residue code
    Journal of The Chinese Institute of Engineers, 2010
    Co-Authors: Hsinchiu Chang, Trieu-kien Truong
    Abstract:

    Abstract An algebraic decoding algorithm is proposed to correct all error patterns of up to three errors in the binary (31, 16, 7) Quadratic Residue (QR) code with reducible generator polynomial. The decoding technique, a modification of the decoding algorithm given by Reed et al., is based on the application of the decoding algorithm proposed by Truong et al. The computation of all syndromes is done in a small field, namely, GF(25). Thus, the computational complexity can be reduced. A full simulation shows that this novel decoding method is superior to the algebraic decoding algorithm given by Reed et al.

  • Algebraic Decoding of the (31, 16, 7) Quadratic Residue Code by Using Berlekamp-Massey Algorithm
    2010 International Conference on Communications and Mobile Computing, 2010
    Co-Authors: Pei-yu Shih, Wen-ku Su, Trieu-kien Truong
    Abstract:

    An algebraic decoding of the (89, 45, 17) Quadratic Residue code suggested by Truong et al. (2008) has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial and applies a verification method to check whether the error pattern obtained by decoding algorithm is correct or not. In this paper, based on the ideas of the algorithm mentioned above, two decoding methods of the (31, 16, 7) binary Quadratic Residue code are proposed. Also, the comparison of the decoding complexity in terms of CPU time of these two methods and the conventional algebraic decoding method proposed by Reed et al. (1990) are given.

Yaotsu Chang - One of the best experts on this subject based on the ideXlab platform.

  • Algebraic Decoding of Some Quadratic Residue Codes With Weak Locators
    IEEE Transactions on Information Theory, 2015
    Co-Authors: Yan-haw Chen, Trieu-kien Truong, Yaotsu Chang
    Abstract:

    In this paper, an explicit expression of the weak-locator polynomial for p-ary Quadratic Residue codes is presented by a modification of the Feng-Tzeng matrix method. The differences between the modified version and the original Feng-Tzeng matrix are that in the new matrix, not every entry is a syndrome, and every syndrome entry is a known syndrome. By utilizing this technique, an algebraic decoding of the ternary (61, 30, 12) Quadratic Residue code is proposed. This new result has never been seen in the literature to our knowledge. An advantage of the proposed decoding algorithm is that in general the obtained weak-locator polynomials can decode efficiently not only all the error patterns of weights four and five, but also some error patterns of weight six.

  • Decoding the (41, 21, 9) Quadratic Residue Code
    2010
    Co-Authors: Yaotsu Chang
    Abstract:

    This paper proposes an algebraic decod- ing algorithm for the (41, 21, 9) Quadratic Residue code via Lagrange interpolation formula to determine error check and error locator polynomials. Programs written in C++ language have been executed to check every possible error pattern of this Quadratic Residue code.

  • decoding binary Quadratic Residue codes using the euclidean algorithm
    Journal of Information Science and Engineering, 2009
    Co-Authors: Pei-yu Shih, Wen-ku Su, Trieu-kien Truong, Yaotsu Chang
    Abstract:

    A simplified algorithm for decoding binary Quadratic Residue (QR) codes is developed in this paper. The key idea is to use the efficient Euclidean algorithm to determine the greatest common divisor of two specific polynomials which can be shown to be the error-locator polynomial. This proposed technique differs from the previous schemes developed for QR codes. It is especially simple due to the well-developed Euclidean algorithm. In this paper, an example using the proposed algorithm to decode the (41, 21, 9) Quadratic Residue code is given and a C++ program of the proposed algorithm has been executed successfully to run all correctable error patterns. The simulations of this new algorithm compared with the Berlekamp-Massey (BM) algorithm for the (71, 36, 11) and (79, 40, 15) Quadratic Residue codes are shown.

  • Syndrome-weight decoder for general binary Quadratic Residue codes
    2009 7th International Conference on Information Communications and Signal Processing (ICICS), 2009
    Co-Authors: Ming-haw Jing, Yaotsu Chang, Zih-heng Chen, Jian-hong Chen
    Abstract:

    Recently, a novel decoding procedure which is called the syndrome-weight determination for the Golay code, or the (23, 12, 7) Quadratic Residue code, was proposed by Chang et al. This method is not only very simple in principle but also suitable for the parallel hardware design. Furthermore, to develop a universal decoding algorithm for arbitrary binary Quadratic Residue codes is very important. In this paper, a simplified decoder for the general binary Quadratic Residue codes of different sizes is developed that is modified by the Chang's decoding scheme. Because of its regular property, the proposed decoder is suitable not only for software design but also for hardware development.

  • algebraic decoding of the 89 45 17 Quadratic Residue code
    IEEE Transactions on Information Theory, 2008
    Co-Authors: Trieu-kien Truong, Pei-yu Shih, Wen-ku Su, Yaotsu Chang
    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.

Wen-ku Su - One of the best experts on this subject based on the ideXlab platform.

  • On the Low-Complexity Soft-Decoding of Binary Quadratic Residue Codes
    Proceedings of the 1st IEEE IIAE International Conference on Intelligent Systems and Image Processing 2013, 2013
    Co-Authors: Wen-ku Su, Pei-yu Shih, Trieu-kien Truong
    Abstract:

    This paper is to shown the performance of a modified algorithm for decoding the (4 7 , 24, 1 1 ) binary Quadratic Residue code up to six errors. The technique in this paper combines the algebraic decoding algorithm with the detection scheme for the (47, 24, 11) Quadratic Residue code offered by Truong et al. to correct up to five errors. Then again, the reliability-search algorithm is utilized to correct one bit of the six bits error. The computer simulation of the scheme shows that at least 9 0 % of 6 bits error occurred are corrected if the E b /N 0 ratio s are greater than 4 dB. As a result, the performance of this modified algorithm is very close to the bound of decoding up to six-bit error.

  • Algebraic Decoding of the (31, 16, 7) Quadratic Residue Code by Using Berlekamp-Massey Algorithm
    2010 International Conference on Communications and Mobile Computing, 2010
    Co-Authors: Pei-yu Shih, Wen-ku Su, Trieu-kien Truong
    Abstract:

    An algebraic decoding of the (89, 45, 17) Quadratic Residue code suggested by Truong et al. (2008) has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial and applies a verification method to check whether the error pattern obtained by decoding algorithm is correct or not. In this paper, based on the ideas of the algorithm mentioned above, two decoding methods of the (31, 16, 7) binary Quadratic Residue code are proposed. Also, the comparison of the decoding complexity in terms of CPU time of these two methods and the conventional algebraic decoding method proposed by Reed et al. (1990) are given.

  • decoding binary Quadratic Residue codes using the euclidean algorithm
    Journal of Information Science and Engineering, 2009
    Co-Authors: Pei-yu Shih, Wen-ku Su, Trieu-kien Truong, Yaotsu Chang
    Abstract:

    A simplified algorithm for decoding binary Quadratic Residue (QR) codes is developed in this paper. The key idea is to use the efficient Euclidean algorithm to determine the greatest common divisor of two specific polynomials which can be shown to be the error-locator polynomial. This proposed technique differs from the previous schemes developed for QR codes. It is especially simple due to the well-developed Euclidean algorithm. In this paper, an example using the proposed algorithm to decode the (41, 21, 9) Quadratic Residue code is given and a C++ program of the proposed algorithm has been executed successfully to run all correctable error patterns. The simulations of this new algorithm compared with the Berlekamp-Massey (BM) algorithm for the (71, 36, 11) and (79, 40, 15) Quadratic Residue codes are shown.

  • On the minimum weights of binary extended Quadratic Residue codes
    2009 11th International Conference on Advanced Communication Technology, 2009
    Co-Authors: Wen-ku Su, Pei-yu Shih, Trieu-kien Truong
    Abstract:

    This paper used an efficient scheme to determine the number of codewords for a given weight in the binary extended Quadratic Residue code. The scheme consists of a weight-counting algorithm and the combinatorial designs of the Assmus-Mattson theorem. As a consequence, the values of minimum weights of the binary (192, 96), (194, 97), and (200, 100) extended Quadratic Residue codes are 28, 28, and 32, respectively. And all the minimum weights of the binary extended Quadratic Residue codes of lengths less than or equal to 200 are determined.

  • A New Scheme to Determine the Weight Distributions of Binary Extended Quadratic Residue Codes
    IEEE Transactions on Communications, 2009
    Co-Authors: Trieu-kien Truong, Y. Chang, Wen-ku Su
    Abstract:

    This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason's theorem to determine the weight distributions of binary extended Quadratic Residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended Quadratic Residue codes are given.

R. Koetter - One of the best experts on this subject based on the ideXlab platform.

J.f. Humphreys - One of the best experts on this subject based on the ideXlab platform.