A New Powerful Scheme Based on Self Invertible Stabilizer Multiplier Permutation to Find the Minimum Distance for large BCH Codes
American Journal of Computer Science and Technology
Volume 1, Issue 2, June 2018, Pages: 39-43
Received: Jan. 7, 2018;
Accepted: Jan. 17, 2018;
Published: Feb. 21, 2018
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Issam Abderrahman Joundan, Faculty of Sciences Ben M'sik, Hassan II University, Casablanca, Morocco
Said Nouh, Faculty of Sciences Ben M'sik, Hassan II University, Casablanca, Morocco
Abdelwahed Namir, Faculty of Sciences Ben M'sik, Hassan II University, Casablanca, Morocco
In this paper, we present the powerful scheme ZSISMP (Zimmermann Self Invertible Stabilizer Multiplier Permutation) to attack the hardness of the minimum distance search problem of BCH codes. This scheme consists in evaluating the minimum distance of the reduced dimension sub code fixed by a Self Invertible Stabilizer Multiplier Permutation by Zimmermann algorithm. The proposed scheme ZSISMP is validated on all BCH codes of known minimum distance. A comparison with several known powerful techniques proves its efficiency in giving more accurate results in short time. The use of this efficient local search had yield to determine the error correcting capability of many BCH codes of length 1023 and 4095.
Issam Abderrahman Joundan,
A New Powerful Scheme Based on Self Invertible Stabilizer Multiplier Permutation to Find the Minimum Distance for large BCH Codes, American Journal of Computer Science and Technology.
Vol. 1, No. 2,
2018, pp. 39-43.
P. Charpin, Open problems on cyclic codes, in: V. S. Pless, W. C. Human, R. A. Brualdi (Eds.), Handbook of Coding Theory, Part 1: Algebraic Coding, Elsevier, Amsterdam, The Netherlands, 1998 (Chapter 11).
C. Ding, X. Du, Z. Zhou, “The Bose and minimum distance of a class of BCH codes,” IEEE Trans. Inf. Theory 61 (5) (2015) 2351–2356.
C. Ding, “Parameters of several classes of BCH codes,” IEEE Trans. Inf. Theory 61 (10) (2015) 5322–5330.
Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar, “A class of non-binary BCH code of Bose and minimum distance.“ Journal of Mathematical and Computational Science (2016), No. 6, 1169-1176.
Cunsheng Ding, Cuiling Fan, Zhengchun Zhou “The dimension and minimum distance of two classes of primitive BCH codes” Finite Fields and Their Applications, Volume 45, May 2017, Pages 237-263.
Hao Liu, Cunsheng Ding, Chengju Li, “Dimensions of three types of BCH codes over GF (q)”, Discrete Mathematics, 340 (2017) 1910–1927.
Shuxing Li “The Minimum Distance of Some Narrow-Sense Primitive BCH Codes”. SIAM Journal on Discrete Mathematics, 2017, Vol. 31, No. 4: pp. 2530-2569.
Shuxing Li, Cunsheng Ding, Maosheng Xiong, and Gennian Ge, “Narrow-Sense BCH Codes Over GF (q) With Length n=qm-1/q-1,” IEEE Transactions on Information Theory (Volume: 63, Issue: 11, Nov. 2017).
Daniel Augot, Pascale Charpin, and Nicolas Sendrier “Studying the Locator Polynomials of Minimum Weight Codewords of BCH Codes” IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 38, NO. 3, MAY 1992.
D. Augot and N. Sendrier, “Idempotents and the BCH bound,” IEEE Trans. Inform. Theory, vol. 40, pp. 204–207, 1994.
A. Canteaut, and F. Chabaud. "A new algorithm for finding minimum-weight words in a linear code: Application to primitive narrow-sense BCH codes of length 511", IEEE Trans. Inform. Theory, IT-44 (1), pp 367-378, Jan. 1998.
J. Stern, “A method for finding codewords of small weight,” in Coding Theory and Applications, G. Cohen and J. Wolfmann, Eds. New York: Springer-Verlag, 1989, pp. 106–113.
Zimmermann K.-H., Integral Hecke Modules, Integral Generalized Reed-Muller Codes, and Linear Codes Technische Universitat HamburgHarburg, Tech. Rep. 3-96, 1996.
The GAP Group. “GAP–Groups, Algorithms, and Programming, Version 4.7.9”. 2015. http://www.gap-system.org.
Grassl, M. Searching for linear codes with large minimum distance. Discovering mathematics with Magma, pp. 287–313, Algorithms Comput. Math., 19, Springer, Berlin, 2006.
M. Zhang, F. Ma, Simulated annealing approach to the minimum distance of error-correcting codes, Int. J. Electron. 76 (1994) 377–384.
J. A. Bland, D. J. Baylis, A tabu search approach to the minimum distance of error-correcting codes, Int. J. Electron. 79 (1995) 829–837.
J. Wallis and K. Houghten, “A Comparative Study of Search Techniques Applied tothe Minumum Distance of BCH Codes,” Conference on Artificial Intelligence and Soft Computing, Banff, 17-19 July 2002.
Askali M., Azouaoui A., Nouh S., Belkasmi M. (2012) On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods, International Journal of Communications, Network and System Sciences, 5 (11), 774-784.
J. A. Bland. Local search optimisation applied to the minimum distance problem. Advanced Engineering Informatics, 21, 2007.
Ajitha Shenoy K. B, Somenath Biswas, Piyush P. Kurur, "Performance of metropolis algorithm for the minimum weight code word problem", Genetic and Evolutionary Computation Conference, 2014.
Bouchaib AYLAJ and Mostafa BELKASMI, “New Simulated Annealing Algorithm for Computing the Minimum Distance of Linear Block Codes”. Advances in Computational Research, indexed Google Scholar ISSN: 0975-3273, E-ISSN: 0975-9085, Volume 6, Issue 1, pp.-153-158, 2014.
C. Berrou, S. Vaton, M. Jezequel and C. Douillard, “Computing the Minimum Distance of Linear Codes by the Error Impulse Method,” Proceedings of IEEE Globecom, Taipei, 17-21 November 2002, pp. 10-14.
S. NOUH, I. A. Joundan, B. Aylaj, M. Belkasmi, A. Namir “New Efficient Scheme Based on Reduction of the Dimension in the Multiple Impulse Method to Find the Minimum Distance of Linear Codes”, International Review on Computers and Software IRECOS, Vol 11, No 9 (2016) pages 742-751.
W. C. Huffman and V. Pless. Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003.
R. C Bose and D. K. Ray-Chaudhuri. On a class of error correcting binary group codes. Information and Control, 3:68–79, mars 1960.
A. Hocquenghem. Codes correcteurs d’erreurs. Chiffres, 2:147–156, sept 1959.