# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020082

## Infinite families of $3$-designs from o-polynomials

 1 Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2 School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China

* Corresponding author: Chunming Tang

Received  November 2019 Revised  February 2020 Published  April 2020

Fund Project: C. Ding's research was supported by the Hong Kong Research Grants Council, Proj. No. 16300415. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds)

A classical approach to constructing combinatorial designs is the group action of a $t$-transitive or $t$-homogeneous permutation group on a base block, which yields a $t$-design in general. It is open how to use a $t$-transitive or $t$-homogeneous permutation group to construct a $(t+1)$-design in general. It is known that the general affine group ${\mathrm{GA}}_1( {\mathrm{GF}}(q))$ is doubly transitive on ${\mathrm{GF}}(q)$. The classical theorem says that the group action by ${\mathrm{GA}}_1( {\mathrm{GF}}(q))$ yields $2$-designs in general. The main objective of this paper is to construct $3$-designs with ${\mathrm{GA}}_1( {\mathrm{GF}}(q))$ and o-polynomials. O-polynomials (equivalently, hyperovals) were used to construct only $2$-designs in the literature. This paper presents for the first time infinite families of $3$-designs from o-polynomials (equivalently, hyperovals).

Citation: Cunsheng Ding, Chunming Tang. Infinite families of $3$-designs from o-polynomials. Advances in Mathematics of Communications, doi: 10.3934/amc.2020082
##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.  Google Scholar [2] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.   Google Scholar [3] W. Cherowitzo, Hyperovals in Desarguesian planes of even order, Combinatorics '86 (Trento, 1986), Annals of Discrete Math., North-Holland, Amsterdam, 37 (1988), 87-94.  doi: 10.1016/S0167-5060(08)70228-0.  Google Scholar [4] W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Disc. Math., 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R.  Google Scholar [5] W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geometriae Dedicata, 60 (1996), 17-37.  doi: 10.1007/BF00150865.  Google Scholar [6] C. S. Ding and C. J. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.  Google Scholar [7] C. S. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combinatorial Theory Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006.  Google Scholar [8] C. Ding and P. Yuan, Five constructions of permutation polynomials over GF$(q^2)$, unpublished manuscript, (2015). http://arXiv.org/abs/1511.00322. Google Scholar [9] C. S. Ding and Z. C. Zhou, Parameters of $2$-designs from some BCH codes, Codes, Cryptography and Information Security, Lecture Notes in Computer Science, Springer, Cham, 10194 (2017), 110-127.  doi: 10.1007/978-3-319-55589-8_8.  Google Scholar [10] D. G. Glynn, Two new sequences of ovals in finite Desarguesian planes of even order, Combinatorial Mathematics X, Lecture Notes in Mathematics, Heidelberg, Springer Verlag, 1983 (1983), 217-229.  doi: 10.1007/BFb0071521.  Google Scholar [11] D. G. Glynn, A condition for the existence of ovals in PG(2, $q$), $q$ even, Geometriae Dedicata, 32 (1989), 247-252.  doi: 10.1007/BF00147433.  Google Scholar [12] W.-A. Jackson, A chracterisation of Hadamard designs with $SL(2, q)$ acting transitively, Geom. Dedicata, 46 (1993), 197-206.  doi: 10.1007/BF01264918.  Google Scholar [13] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar [14] A. Maschietti, Difference set and hyperovals, Des. Codes Cryptg., 14 (1998), 89-98.  doi: 10.1023/A:1008264606494.  Google Scholar [15] S. E. Payne, A new infinite family of generalized quadrangles, Congressus Numerantium, 49 (1985), 115-128.   Google Scholar [16] B. Segre, Sui $k$-archi nei piani finiti di caratteristica 2, Revue de Math. Pures Appl., 2 (1957), 289-300.   Google Scholar [17] B. Segre, Ovali e curvenei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend., 32 (1962), 785-790.   Google Scholar [18] B. Segre and U. Bartocci, Ovali ed alte curve nei piani di Galois di caratteristica due, Acta Arith., 18 (1971), 423-449.  doi: 10.4064/aa-18-1-423-449.  Google Scholar [19] N. V. Semakov and V. A. Zinov'ev, Balanced codes and tactical configurations, Problemy Peredachi Informatsii, 5 (1969), 22-28.   Google Scholar [20] M. S. Shrikhande, Quasi-symmetric designs, Handbook of Combinatorial Designs, 2nd Edition, CRC Press, New York, (2007), 578–582. Google Scholar [21] C. M. Tang, Infinite families of 3-designs from APN functions, J. Combinatorial Designs, 28 (2020), 97-117.  doi: 10.1002/jcd.21685.  Google Scholar [22] V. D. Tonchev, Codes and designs, Handbook of coding theory, North-Holland, Amsterdam, 1, 2 (1998), 1229-1267.   Google Scholar [23] Q. Xiang, On balanced binary sequences with two-level autocorrelation functions, IEEE Trans. Inf. Theory, 44 (1998), 3153-3156.  doi: 10.1109/18.737547.  Google Scholar

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##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.  Google Scholar [2] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.   Google Scholar [3] W. Cherowitzo, Hyperovals in Desarguesian planes of even order, Combinatorics '86 (Trento, 1986), Annals of Discrete Math., North-Holland, Amsterdam, 37 (1988), 87-94.  doi: 10.1016/S0167-5060(08)70228-0.  Google Scholar [4] W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Disc. Math., 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R.  Google Scholar [5] W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geometriae Dedicata, 60 (1996), 17-37.  doi: 10.1007/BF00150865.  Google Scholar [6] C. S. Ding and C. J. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.  Google Scholar [7] C. S. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combinatorial Theory Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006.  Google Scholar [8] C. Ding and P. Yuan, Five constructions of permutation polynomials over GF$(q^2)$, unpublished manuscript, (2015). http://arXiv.org/abs/1511.00322. Google Scholar [9] C. S. Ding and Z. C. Zhou, Parameters of $2$-designs from some BCH codes, Codes, Cryptography and Information Security, Lecture Notes in Computer Science, Springer, Cham, 10194 (2017), 110-127.  doi: 10.1007/978-3-319-55589-8_8.  Google Scholar [10] D. G. Glynn, Two new sequences of ovals in finite Desarguesian planes of even order, Combinatorial Mathematics X, Lecture Notes in Mathematics, Heidelberg, Springer Verlag, 1983 (1983), 217-229.  doi: 10.1007/BFb0071521.  Google Scholar [11] D. G. Glynn, A condition for the existence of ovals in PG(2, $q$), $q$ even, Geometriae Dedicata, 32 (1989), 247-252.  doi: 10.1007/BF00147433.  Google Scholar [12] W.-A. Jackson, A chracterisation of Hadamard designs with $SL(2, q)$ acting transitively, Geom. Dedicata, 46 (1993), 197-206.  doi: 10.1007/BF01264918.  Google Scholar [13] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar [14] A. Maschietti, Difference set and hyperovals, Des. Codes Cryptg., 14 (1998), 89-98.  doi: 10.1023/A:1008264606494.  Google Scholar [15] S. E. Payne, A new infinite family of generalized quadrangles, Congressus Numerantium, 49 (1985), 115-128.   Google Scholar [16] B. Segre, Sui $k$-archi nei piani finiti di caratteristica 2, Revue de Math. Pures Appl., 2 (1957), 289-300.   Google Scholar [17] B. Segre, Ovali e curvenei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend., 32 (1962), 785-790.   Google Scholar [18] B. Segre and U. Bartocci, Ovali ed alte curve nei piani di Galois di caratteristica due, Acta Arith., 18 (1971), 423-449.  doi: 10.4064/aa-18-1-423-449.  Google Scholar [19] N. V. Semakov and V. A. Zinov'ev, Balanced codes and tactical configurations, Problemy Peredachi Informatsii, 5 (1969), 22-28.   Google Scholar [20] M. S. Shrikhande, Quasi-symmetric designs, Handbook of Combinatorial Designs, 2nd Edition, CRC Press, New York, (2007), 578–582. Google Scholar [21] C. M. Tang, Infinite families of 3-designs from APN functions, J. Combinatorial Designs, 28 (2020), 97-117.  doi: 10.1002/jcd.21685.  Google Scholar [22] V. D. Tonchev, Codes and designs, Handbook of coding theory, North-Holland, Amsterdam, 1, 2 (1998), 1229-1267.   Google Scholar [23] Q. Xiang, On balanced binary sequences with two-level autocorrelation functions, IEEE Trans. Inf. Theory, 44 (1998), 3153-3156.  doi: 10.1109/18.737547.  Google Scholar
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