American Institute of Mathematical Sciences

Advanced
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

[1]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[2]

Martino Borello, Olivier Mila. Symmetries of weight enumerators and applications to Reed-Muller codes. Advances in Mathematics of Communications, 2019, 13 (2) : 313-328. doi: 10.3934/amc.2019021
[3]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[4]

Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333
[5]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[6]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[7]

Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010
[8]

Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041
[9]

Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007
[10]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[11]

María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018
[12]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[13]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[14]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[15]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002
[16]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[17]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[18]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[19]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[20]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (173)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences