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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 |
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 $-designs from o-polynomials (equivalently, hyperovals).
References:
[1] |
E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
![]() ![]() |
[3] |
W. Cherowitzo,
Hyperovals in Desarguesian planes of even order, Combinatorics '86 (Trento, 1986), Annals of Discrete Math., North-Holland, Amsterdam, 37 (1988), 87-94.
doi: 10.1016/S0167-5060(08)70228-0. |
[4] |
W. Cherowitzo,
Hyperovals in Desarguesian planes: An update, Disc. Math., 155 (1996), 31-38.
doi: 10.1016/0012-365X(94)00367-R. |
[5] |
W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle,
Flocks and ovals, Geometriae Dedicata, 60 (1996), 17-37.
doi: 10.1007/BF00150865. |
[6] |
C. S. Ding and C. J. Li,
Infinite families of 2-designs and 3-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.
doi: 10.1016/j.disc.2017.05.013. |
[7] |
C. S. Ding and J. Yuan,
A family of skew Hadamard difference sets, J. Combinatorial Theory Ser. A, 113 (2006), 1526-1535.
doi: 10.1016/j.jcta.2005.10.006. |
[8] |
C. Ding and P. Yuan, Five constructions of permutation polynomials over GF$(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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