Infinite families of <inline-formula><tex-math id="M1">\begin{document} 3 \end{document}</tex-math></inline-formula>-designs from o-polynomials

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November 2021, 15(4): 557-573. doi: 10.3934/amc.2020082

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

Cunsheng Ding ^1, and Chunming Tang ^2,,

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 November 2021 Early access 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).

Keywords: Hyperoval, o-polynomial, polynomial, projective plane, t-design.

Mathematics Subject Classification: Primary: 51E21, 05B05, 12E10.

Citation: Cunsheng Ding, Chunming Tang. Infinite families of $ 3 $-designs from o-polynomials. Advances in Mathematics of Communications, 2021, 15 (4) : 557-573. 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.
[2]	T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
[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.
[4]	W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Disc. Math., 155 (1996), 31-38. doi: 10.1016/0012-365X(94)00367-R.
[5]	W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geometriae Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.
[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.
[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.
[8]	C. Ding and P. Yuan, Five constructions of permutation polynomials over GF$(q^2)$, unpublished manuscript, (2015). http://arXiv.org/abs/1511.00322.
[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.
[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.
[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.
[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.
[13]	R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
[14]	A. Maschietti, Difference set and hyperovals, Des. Codes Cryptg., 14 (1998), 89-98. doi: 10.1023/A:1008264606494.
[15]	S. E. Payne, A new infinite family of generalized quadrangles, Congressus Numerantium, 49 (1985), 115-128.
[16]	B. Segre, Sui $k$-archi nei piani finiti di caratteristica 2, Revue de Math. Pures Appl., 2 (1957), 289-300.
[17]	B. Segre, Ovali e curvenei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend., 32 (1962), 785-790.
[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.
[19]	N. V. Semakov and V. A. Zinov'ev, Balanced codes and tactical configurations, Problemy Peredachi Informatsii, 5 (1969), 22-28.
[20]	M. S. Shrikhande, Quasi-symmetric designs, Handbook of Combinatorial Designs, 2nd Edition, CRC Press, New York, (2007), 578–582.
[21]	C. M. Tang, Infinite families of 3-designs from APN functions, J. Combinatorial Designs, 28 (2020), 97-117. doi: 10.1002/jcd.21685.
[22]	V. D. Tonchev, Codes and designs, Handbook of coding theory, North-Holland, Amsterdam, 1, 2 (1998), 1229-1267.
[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.

show all references

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.
[2]	T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
[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.
[4]	W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Disc. Math., 155 (1996), 31-38. doi: 10.1016/0012-365X(94)00367-R.
[5]	W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geometriae Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.
[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.
[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.
[8]	C. Ding and P. Yuan, Five constructions of permutation polynomials over GF$(q^2)$, unpublished manuscript, (2015). http://arXiv.org/abs/1511.00322.
[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.
[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.
[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.
[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.
[13]	R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
[14]	A. Maschietti, Difference set and hyperovals, Des. Codes Cryptg., 14 (1998), 89-98. doi: 10.1023/A:1008264606494.
[15]	S. E. Payne, A new infinite family of generalized quadrangles, Congressus Numerantium, 49 (1985), 115-128.
[16]	B. Segre, Sui $k$-archi nei piani finiti di caratteristica 2, Revue de Math. Pures Appl., 2 (1957), 289-300.
[17]	B. Segre, Ovali e curvenei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend., 32 (1962), 785-790.
[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.
[19]	N. V. Semakov and V. A. Zinov'ev, Balanced codes and tactical configurations, Problemy Peredachi Informatsii, 5 (1969), 22-28.
[20]	M. S. Shrikhande, Quasi-symmetric designs, Handbook of Combinatorial Designs, 2nd Edition, CRC Press, New York, (2007), 578–582.
[21]	C. M. Tang, Infinite families of 3-designs from APN functions, J. Combinatorial Designs, 28 (2020), 97-117. doi: 10.1002/jcd.21685.
[22]	V. D. Tonchev, Codes and designs, Handbook of coding theory, North-Holland, Amsterdam, 1, 2 (1998), 1229-1267.
[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.

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