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A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane
A spectral characterisation of $ t $-designs and its applications
1. | Department of Mathematics, Pusan National University, Republic of Korea |
2. | Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China |
3. | Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea |
There are two standard approaches to the construction of $ t $-designs. The first one is based on permutation group actions on certain base blocks. The second one is based on coding theory. The objective of this paper is to give a spectral characterisation of all $ t $-designs by introducing a characteristic Boolean function of a $ t $-design. The spectra of the characteristic functions of $ (n-2)/2 $-$ (n, n/2, 1) $ Steiner systems are determined and properties of such designs are proved. Delsarte's characterisations of orthogonal arrays and $ t $-designs, which are two special cases of Delsarte's characterisation of $ T $-designs in association schemes, are slightly extended into two spectral characterisations. Another characterisation of $ t $-designs by Delsarte and Seidel is also extended into a spectral one. These spectral characterisations are then compared with the new spectral characterisation of this paper.
References:
[1] |
W. O. Alltop,
Extending t-designs, J. Comb. Theory A, 18 (1975), 177-186.
doi: 10.1016/0097-3165(75)90006-0. |
[2] |
E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836.![]() ![]() |
[3] |
A. H. Baartmans, I. Bluskov and V. D. Tonchev,
The Preparata codes and a class of 4-designs, J. Combinatorial Designs, 2 (1994), 167-170.
doi: 10.1002/jcd.3180020307. |
[4] |
T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
![]() |
[5] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin Heidelberg, 1989.
doi: 10.1007/978-3-642-74341-2. |
[6] |
C. Carlet and C. Ding,
Nonlinearities of S-boxes, Finite Fields and Their Applications, 13 (2007), 121-135.
doi: 10.1016/j.ffa.2005.07.003. |
[7] |
S. Chang and J. Y. Hyun,
Linear codes from simplicial complexes, Des. Codes Cryptogr., 86 (2018), 2167-2181.
doi: 10.1007/s10623-017-0442-5. |
[8] |
C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, 2nd edition, CRC Press, New York, 2007. |
[9] |
P. Delsarte,
An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 1-97.
|
[10] |
P. Delsarte,
Pairs of vectors in the space of an association scheme, Philips Res. Rep., 32 (1977), 373-411.
|
[11] |
P. Delsarte and J. J. Seidel,
Fisher type inequalities for Euclidean t-designs, Linear Algebra and Its Applications, 114/115 (1989), 213-230.
doi: 10.1016/0024-3795(89)90462-X. |
[12] |
C. Ding,
A construction of binary linear codes from Boolean functions, Discrete Mathematics, 339 (2016), 2288-2303.
doi: 10.1016/j.disc.2016.03.029. |
[13] |
C. Ding and Z. Zhou, Parameters of 2-designs from some BCH codes, in: Codes, Cryptography and Information Security (eds. S. El Hajji, A. Nitaj and E. M. Souidi), Lecture Notes in Computer Science, Vol. 10194, Springer, (2017), 110–127. |
[14] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() |
[15] |
P. Keevash, The existence of designs, arXiv: 1401.3665v2 [math.CO]. Google Scholar |
[16] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[17] |
L. Teirlinck,
Nontrivial t-designs without repeated blocks exist for all t, Discrete Math., 65 (1987), 301-311.
doi: 10.1016/0012-365X(87)90061-6. |
[18] |
V. D. Tonchev,
A class of Steiner 4-wise balanced designs derived from Preparata codes, J. Combinatorial Designs, 4 (1996), 203-204.
doi: 10.1002/(SICI)1520-6610(1996)4:3<203::AID-JCD3>3.0.CO;2-J. |
[19] |
V. D. Tonchev, Codes and designs, in: Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), Vol. Ⅱ, Elsevier, Amsterdam, (1998), 1229–1267. |
[20] |
V. D. Tonchev, Codes, in Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), 2nd edition, CRC Press, New York, (2007), 677–701. Google Scholar |
[21] |
T. Wadayama, T. Hada, K. Wakasugi and M. Kasahara,
Upper and lower bounds on maximum nonlinearity of n-input m-output Boolean function, Designs, Codes Cryptography, 23 (2001), 23-33.
doi: 10.1023/A:1011207501748. |
show all references
References:
[1] |
W. O. Alltop,
Extending t-designs, J. Comb. Theory A, 18 (1975), 177-186.
doi: 10.1016/0097-3165(75)90006-0. |
[2] |
E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836.![]() ![]() |
[3] |
A. H. Baartmans, I. Bluskov and V. D. Tonchev,
The Preparata codes and a class of 4-designs, J. Combinatorial Designs, 2 (1994), 167-170.
doi: 10.1002/jcd.3180020307. |
[4] |
T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
![]() |
[5] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin Heidelberg, 1989.
doi: 10.1007/978-3-642-74341-2. |
[6] |
C. Carlet and C. Ding,
Nonlinearities of S-boxes, Finite Fields and Their Applications, 13 (2007), 121-135.
doi: 10.1016/j.ffa.2005.07.003. |
[7] |
S. Chang and J. Y. Hyun,
Linear codes from simplicial complexes, Des. Codes Cryptogr., 86 (2018), 2167-2181.
doi: 10.1007/s10623-017-0442-5. |
[8] |
C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, 2nd edition, CRC Press, New York, 2007. |
[9] |
P. Delsarte,
An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 1-97.
|
[10] |
P. Delsarte,
Pairs of vectors in the space of an association scheme, Philips Res. Rep., 32 (1977), 373-411.
|
[11] |
P. Delsarte and J. J. Seidel,
Fisher type inequalities for Euclidean t-designs, Linear Algebra and Its Applications, 114/115 (1989), 213-230.
doi: 10.1016/0024-3795(89)90462-X. |
[12] |
C. Ding,
A construction of binary linear codes from Boolean functions, Discrete Mathematics, 339 (2016), 2288-2303.
doi: 10.1016/j.disc.2016.03.029. |
[13] |
C. Ding and Z. Zhou, Parameters of 2-designs from some BCH codes, in: Codes, Cryptography and Information Security (eds. S. El Hajji, A. Nitaj and E. M. Souidi), Lecture Notes in Computer Science, Vol. 10194, Springer, (2017), 110–127. |
[14] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() |
[15] |
P. Keevash, The existence of designs, arXiv: 1401.3665v2 [math.CO]. Google Scholar |
[16] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[17] |
L. Teirlinck,
Nontrivial t-designs without repeated blocks exist for all t, Discrete Math., 65 (1987), 301-311.
doi: 10.1016/0012-365X(87)90061-6. |
[18] |
V. D. Tonchev,
A class of Steiner 4-wise balanced designs derived from Preparata codes, J. Combinatorial Designs, 4 (1996), 203-204.
doi: 10.1002/(SICI)1520-6610(1996)4:3<203::AID-JCD3>3.0.CO;2-J. |
[19] |
V. D. Tonchev, Codes and designs, in: Handbook of Coding Theory (eds. V. S. Pless and W. C. Huffman), Vol. Ⅱ, Elsevier, Amsterdam, (1998), 1229–1267. |
[20] |
V. D. Tonchev, Codes, in Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), 2nd edition, CRC Press, New York, (2007), 677–701. Google Scholar |
[21] |
T. Wadayama, T. Hada, K. Wakasugi and M. Kasahara,
Upper and lower bounds on maximum nonlinearity of n-input m-output Boolean function, Designs, Codes Cryptography, 23 (2001), 23-33.
doi: 10.1023/A:1011207501748. |
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