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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

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., Univ. Sydney, 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), 93-101. 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), 231-239. 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), 175-186. 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), 26-53. 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), 403-442.  Google Scholar

[8]

J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes, IEEE Trans. Inform. Theory, 14 (1968), 182-188.  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), 153-226.  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), 285-297. 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), 131-140. 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, Amsterdam, 1977. Google Scholar

[13]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.  Google Scholar

[14]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 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), 38-49.  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), 627-628. Google Scholar

[18]

V. D. Tonchev, Combinatorial Configurations, Longman-Wiley, New York, 1988.  Google Scholar

[19]

E. J. Weldon, Euclidean geometry cyclic codes, in Proc. Conf. Combin. Math. Appl., Univ. North Carolina, 1967. Google Scholar

show all references

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., Univ. Sydney, 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), 93-101. 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), 231-239. 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), 175-186. 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), 26-53. 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), 403-442.  Google Scholar

[8]

J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes, IEEE Trans. Inform. Theory, 14 (1968), 182-188.  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), 153-226.  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), 285-297. 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), 131-140. 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, Amsterdam, 1977. Google Scholar

[13]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.  Google Scholar

[14]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 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), 38-49.  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), 627-628. Google Scholar

[18]

V. D. Tonchev, Combinatorial Configurations, Longman-Wiley, New York, 1988.  Google Scholar

[19]

E. J. Weldon, Euclidean geometry cyclic codes, in Proc. Conf. Combin. Math. Appl., Univ. North Carolina, 1967. Google Scholar

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