The weight distribution of the self-dual [128,64] polarity design code

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August 2016, 10(3): 643-648. doi: 10.3934/amc.2016032

The weight distribution of the self-dual $[128,64]$ polarity design code

Masaaki Harada ^1, , Ethan Novak ^2, and Vladimir D. Tonchev ^2,

1.	Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
2.	Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, United States, United States

Received July 2015 Revised February 2016 Published August 2016

Abstract
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The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.

Keywords: Reed-Muller code, polarity design., Self-dual code, weight distribution.

Mathematics Subject Classification: Primary: 94B30; Secondary: 05B2.

Citation: Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032

References:

[1]	E. F. Assmus, Jr. and J. D. Key, Designs and their Codes,, Cambridge Univ. Press, (1992). doi: 10.1017/CBO9781316529836. Google Scholar
[2]	W. Bosma and J. Cannon, Handbook of Magma Functions,, Dep. Math., (1994). Google Scholar
[3]	N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length $64$ through neighbors and covering radii,, Des. Codes Cryptogr., 42 (2007), 93. doi: 10.1007/s10623-006-9018-5. Google Scholar
[4]	D. Clark, D. Jungnickel and V. D. Tonchev, Affine geometry designs, polarities, and Hamada's conjecture,, J. Combin. Theory Ser. A, 118 (2011), 231. doi: 10.1016/j.jcta.2010.06.007. Google Scholar
[5]	D. Clark and V. D. Tonchev, A new class of majority-logic decodable codes derived from polarity designs,, Adv. Math. Commun., 7 (2013), 175. doi: 10.3934/amc.2013.7.175. Google Scholar
[6]	J. H. Conway and V. Pless, On the enumeration of self-dual codes,, J. Combin. Theory Ser. A, 28 (1980), 26. doi: 10.1016/0097-3165(80)90057-6. Google Scholar
[7]	P. Delsarte, J.-M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives,, Inform. Control, 16 (1970), 403. Google Scholar
[8]	J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes,, IEEE Trans. Inform. Theory, 14 (1968), 182. Google Scholar
[9]	N. Hamada, On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error-correcting codes,, Hiroshima Math. J., 3 (1973), 153. Google Scholar
[10]	N. Hamada, On the geometric structure and $p$-rank of affine triple system derived from a nonassociative Moufang Loop with the maximum associative center,, J. Combin. Theory Ser. A, 30 (1981), 285. doi: 10.1016/0097-3165(81)90024-8. Google Scholar
[11]	D. Jungnickel and V. D. Tonchev, Polarities, quasi-symmetric designs, and Hamada's conjecture,, Des. Codes Cryptogr., 51 (2009), 131. doi: 10.1007/s10623-008-9249-8. Google Scholar
[12]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1977). Google Scholar
[13]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes,, Inform. Control, 22 (1973), 188. Google Scholar
[14]	G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory,, Springer, (2006). Google Scholar
[15]	The OEIS Foundation (founded by N.J.A. Sloane in 1964), The Online Encyclopedia of Integer Sequences,, , (). Google Scholar
[16]	I. S. Reed, A class of multiple-error correcting codes and the decoding scheme,, IRE Trans. Inform. Theory, 4 (1954), 38. Google Scholar
[17]	M. Sugino, Y. Ienaga, M. Tokura and T. Kasami, Weight distribution of (128,64) Reed-Muller code,, IEEE Trans. Inform. Theory, 17 (1971), 627. Google Scholar
[18]	V. D. Tonchev, Combinatorial Configurations,, Longman-Wiley, (1988). Google Scholar
[19]	E. J. Weldon, Euclidean geometry cyclic codes,, in Proc. Conf. Combin. Math. Appl., (1967). Google Scholar

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References:

[1]	E. F. Assmus, Jr. and J. D. Key, Designs and their Codes,, Cambridge Univ. Press, (1992). doi: 10.1017/CBO9781316529836. Google Scholar
[2]	W. Bosma and J. Cannon, Handbook of Magma Functions,, Dep. Math., (1994). Google Scholar
[3]	N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length $64$ through neighbors and covering radii,, Des. Codes Cryptogr., 42 (2007), 93. doi: 10.1007/s10623-006-9018-5. Google Scholar
[4]	D. Clark, D. Jungnickel and V. D. Tonchev, Affine geometry designs, polarities, and Hamada's conjecture,, J. Combin. Theory Ser. A, 118 (2011), 231. doi: 10.1016/j.jcta.2010.06.007. Google Scholar
[5]	D. Clark and V. D. Tonchev, A new class of majority-logic decodable codes derived from polarity designs,, Adv. Math. Commun., 7 (2013), 175. doi: 10.3934/amc.2013.7.175. Google Scholar
[6]	J. H. Conway and V. Pless, On the enumeration of self-dual codes,, J. Combin. Theory Ser. A, 28 (1980), 26. doi: 10.1016/0097-3165(80)90057-6. Google Scholar
[7]	P. Delsarte, J.-M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives,, Inform. Control, 16 (1970), 403. Google Scholar
[8]	J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes,, IEEE Trans. Inform. Theory, 14 (1968), 182. Google Scholar
[9]	N. Hamada, On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error-correcting codes,, Hiroshima Math. J., 3 (1973), 153. Google Scholar
[10]	N. Hamada, On the geometric structure and $p$-rank of affine triple system derived from a nonassociative Moufang Loop with the maximum associative center,, J. Combin. Theory Ser. A, 30 (1981), 285. doi: 10.1016/0097-3165(81)90024-8. Google Scholar
[11]	D. Jungnickel and V. D. Tonchev, Polarities, quasi-symmetric designs, and Hamada's conjecture,, Des. Codes Cryptogr., 51 (2009), 131. doi: 10.1007/s10623-008-9249-8. Google Scholar
[12]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1977). Google Scholar
[13]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes,, Inform. Control, 22 (1973), 188. Google Scholar
[14]	G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory,, Springer, (2006). Google Scholar
[15]	The OEIS Foundation (founded by N.J.A. Sloane in 1964), The Online Encyclopedia of Integer Sequences,, , (). Google Scholar
[16]	I. S. Reed, A class of multiple-error correcting codes and the decoding scheme,, IRE Trans. Inform. Theory, 4 (1954), 38. Google Scholar
[17]	M. Sugino, Y. Ienaga, M. Tokura and T. Kasami, Weight distribution of (128,64) Reed-Muller code,, IEEE Trans. Inform. Theory, 17 (1971), 627. Google Scholar
[18]	V. D. Tonchev, Combinatorial Configurations,, Longman-Wiley, (1988). Google Scholar
[19]	E. J. Weldon, Euclidean geometry cyclic codes,, in Proc. Conf. Combin. Math. Appl., (1967). Google Scholar

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