# American Institute of Mathematical Sciences

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

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

 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 $d \in \{ 20, 24 \}$.
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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