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August  2019, 13(3): 477-503. doi: 10.3934/amc.2019030

A spectral characterisation of $ t $-designs and its applications

Eun-Kyung Cho 1, , Cunsheng Ding 2,, and  Jong Yoon Hyun 3,

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

* Corresponding author

Received  August 2018 Published  April 2019

Fund Project: C. Ding was supported by The Hong Kong Grants Council, Proj. No. 16300418. J. Y. Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1D1A1B05030707).

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.

Keywords: Association scheme, Boolean function, code, Steiner system, t-design, Walsh transform.
Mathematics Subject Classification: Primary: 05B05, 06E30; Secondary: 51E22.
Citation: Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030
References:
[1]

W. O. Alltop, Extending t-designs, J. Comb. Theory A, 18 (1975), 177-186.  doi: 10.1016/0097-3165(75)90006-0.  Google Scholar

[2] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[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.  Google Scholar

[4] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, 2nd edition, CRC Press, New York, 2007.  Google Scholar

[9]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 1-97.   Google Scholar

[10]

P. Delsarte, Pairs of vectors in the space of an association scheme, Philips Res. Rep., 32 (1977), 373-411.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[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.  Google Scholar

[4] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, 2nd edition, CRC Press, New York, 2007.  Google Scholar

[9]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 1-97.   Google Scholar

[10]

P. Delsarte, Pairs of vectors in the space of an association scheme, Philips Res. Rep., 32 (1977), 373-411.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  Spectrum of $ f_{ {\mathbb{D}}} $
Weight of $ w $ Multiset $ \{ \hat{f}_{ {\mathbb{D}}}(w) \} $
$ 0, 12 $ $ \{ 132 \} $
$ 1, 11 $ $ \{ 0^{12} \} $
$ 2, 10 $ $ \{ -12^{66} \} $
$ 3, 9 $ $ \{ 0^{220} \} $
$ 4, 8 $ $ \{ 4^{495} \} $
$ 5, 7 $ $ \{ 0^{792} \} $
$ 6 $ $ \{ -12^{792}, 52^{132}\} $
Table Options

Download as excel

Weight of $ w $ Multiset $ \{ \hat{f}_{ {\mathbb{D}}}(w) \} $
$ 0, 12 $ $ \{ 132 \} $
$ 1, 11 $ $ \{ 0^{12} \} $
$ 2, 10 $ $ \{ -12^{66} \} $
$ 3, 9 $ $ \{ 0^{220} \} $
$ 4, 8 $ $ \{ 4^{495} \} $
$ 5, 7 $ $ \{ 0^{792} \} $
$ 6 $ $ \{ -12^{792}, 52^{132}\} $
Table 2.  Weight distribution
Weight $ w $ No. of codewords $ A_w $
$ 0 $ $ 1 $
$ 132 $ $ 1 $
$ 2^{11}-12 $ $ 924 $
$ 2^{11} $ $ 6143 $
$ 2^{11}+4 $ $ 990 $
$ 2^{11}+52 $ $ 132 $
$ 2^{11}+132 $ $ 1 $
Table Options

Download as excel

Weight $ w $ No. of codewords $ A_w $
$ 0 $ $ 1 $
$ 132 $ $ 1 $
$ 2^{11}-12 $ $ 924 $
$ 2^{11} $ $ 6143 $
$ 2^{11}+4 $ $ 990 $
$ 2^{11}+52 $ $ 132 $
$ 2^{11}+132 $ $ 1 $
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