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On self-dual MRD codes
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 |
References:
[1] |
E. F. Assmus, Jr. and J. D. Key, Designs and their Codes, Cambridge Univ. Press, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
W. Bosma and J. Cannon, Handbook of Magma Functions, Dep. Math., Univ. Sydney, 1994. |
[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. |
[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. |
[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. |
[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. |
[7] |
P. Delsarte, J.-M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Inform. Control, 16 (1970), 403-442. |
[8] |
J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes, IEEE Trans. Inform. Theory, 14 (1968), 182-188. |
[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. |
[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. |
[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. |
[12] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[13] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. |
[14] |
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006. |
[15] |
The OEIS Foundation (founded by N.J.A. Sloane in 1964), The Online Encyclopedia of Integer Sequences,, , ().
|
[16] |
I. S. Reed, A class of multiple-error correcting codes and the decoding scheme, IRE Trans. Inform. Theory, 4 (1954), 38-49. |
[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. |
[18] |
V. D. Tonchev, Combinatorial Configurations, Longman-Wiley, New York, 1988. |
[19] |
E. J. Weldon, Euclidean geometry cyclic codes, in Proc. Conf. Combin. Math. Appl., Univ. North Carolina, 1967. |
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. |
[2] |
W. Bosma and J. Cannon, Handbook of Magma Functions, Dep. Math., Univ. Sydney, 1994. |
[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. |
[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. |
[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. |
[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. |
[7] |
P. Delsarte, J.-M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Inform. Control, 16 (1970), 403-442. |
[8] |
J.-M. Goethals and P. Delsarte, On a class of majority-decodable cyclic codes, IEEE Trans. Inform. Theory, 14 (1968), 182-188. |
[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. |
[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. |
[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. |
[12] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[13] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. |
[14] |
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006. |
[15] |
The OEIS Foundation (founded by N.J.A. Sloane in 1964), The Online Encyclopedia of Integer Sequences,, , ().
|
[16] |
I. S. Reed, A class of multiple-error correcting codes and the decoding scheme, IRE Trans. Inform. Theory, 4 (1954), 38-49. |
[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. |
[18] |
V. D. Tonchev, Combinatorial Configurations, Longman-Wiley, New York, 1988. |
[19] |
E. J. Weldon, Euclidean geometry cyclic codes, in Proc. Conf. Combin. Math. Appl., Univ. North Carolina, 1967. |
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