# American Institute of Mathematical Sciences

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

## 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

* 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.

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