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Erschienen in: Designs, Codes and Cryptography 3/2015

01.03.2015

Paley type sets from cyclotomic classes and Arasu–Dillon–Player difference sets

verfasst von: Yu Qing Chen, Tao Feng

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2015

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Abstract

In this paper, we present constructions of abelian Paley type sets by using multiplicative characters of finite fields and Arasu–Dillon–Player difference sets. The constructions produce many new Paley type sets and their configurations that were previous unknown in our classification of Paley type sets in finite fields of small orders.
Literatur
2.
Zurück zum Zitat Arasu K.T.: A reduction theorem for circulant weighing matrices. Australas. J. Comb. 18, 111–114 (1998). Arasu K.T.: A reduction theorem for circulant weighing matrices. Australas. J. Comb. 18, 111–114 (1998).
3.
Zurück zum Zitat Arasu K.T., Chen Y.Q., Dillon J.F., Liu X., Player K.J.: Abelian difference sets of order n dividing \(\lambda \). Des. Codes Cryptogr. 44, 307–319 (2007). Arasu K.T., Chen Y.Q., Dillon J.F., Liu X., Player K.J.: Abelian difference sets of order n dividing \(\lambda \). Des. Codes Cryptogr. 44, 307–319 (2007).
4.
Zurück zum Zitat Arasu K.T., Dillon J.F., Player K.J.: Character sum factorizations yield perfect sequences (in press). Arasu K.T., Dillon J.F., Player K.J.: Character sum factorizations yield perfect sequences (in press).
5.
Zurück zum Zitat Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Determination of all possible orders of weight 16 circulant weighing matrices. Finite Fields Appl. 12, 498–538 (2006). Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Determination of all possible orders of weight 16 circulant weighing matrices. Finite Fields Appl. 12, 498–538 (2006).
6.
Zurück zum Zitat Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Circulant weighing matrices of weight \(2^{2t}\). Des. Codes Cryptogr. 41, 111–123 (2006). Arasu K.T., Leung K.H., Ma S.L., Nabavi A., Ray-Chaudhuri D.K.: Circulant weighing matrices of weight \(2^{2t}\). Des. Codes Cryptogr. 41, 111–123 (2006).
7.
Zurück zum Zitat Arasu K.T., Ma S.L.: Some new results on circulant weighing matrices. J. Algebraic Comb. 14, 91–101 (2001). Arasu K.T., Ma S.L.: Some new results on circulant weighing matrices. J. Algebraic Comb. 14, 91–101 (2001).
8.
Zurück zum Zitat Berndt B.C., Evans R.J., Williams K.S.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1998). Berndt B.C., Evans R.J., Williams K.S.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1998).
9.
Zurück zum Zitat Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999). Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999).
10.
Zurück zum Zitat Carlitz L.: A theorem on permutations in a finite field. Proc. Am. Math. Soc. 11, 456–459 (1960). Carlitz L.: A theorem on permutations in a finite field. Proc. Am. Math. Soc. 11, 456–459 (1960).
11.
Zurück zum Zitat Camion P., Mann H.B.: Antisymmetric difference sets. J. Number Theory 4, 266–268 (1972). Camion P., Mann H.B.: Antisymmetric difference sets. J. Number Theory 4, 266–268 (1972).
12.
Zurück zum Zitat Chen Y.Q.: On the existence of abelian Hadamard difference sets and a new family of difference sets. Finite Fields Appl. 3, 234–256 (1997). Chen Y.Q.: On the existence of abelian Hadamard difference sets and a new family of difference sets. Finite Fields Appl. 3, 234–256 (1997).
13.
Zurück zum Zitat Chen Y.Q.: Multiplicative characterization of some difference sets in elementary abelian groups. J. Comb. Inf. Syst. Sci. 34, 95–111 (2009). Chen Y.Q.: Multiplicative characterization of some difference sets in elementary abelian groups. J. Comb. Inf. Syst. Sci. 34, 95–111 (2009).
14.
Zurück zum Zitat Chen Y.Q.: Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles. J. Comb. Theory Ser. A 118, 2185–2206 (2011). Chen Y.Q.: Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles. J. Comb. Theory Ser. A 118, 2185–2206 (2011).
15.
Zurück zum Zitat Chen Y.Q., Feng T.: Abelian and non-abelian Paley type group schemes. Des. Codes Cryptogr. 68, 141–154 (2013). Chen Y.Q., Feng T.: Abelian and non-abelian Paley type group schemes. Des. Codes Cryptogr. 68, 141–154 (2013).
16.
Zurück zum Zitat Chen Y.Q., Polhill J.: Paley type group schemes and planar Dembowski–Ostrom polynomials. Discret. Math. 311, 1349–1364 (2011). Chen Y.Q., Polhill J.: Paley type group schemes and planar Dembowski–Ostrom polynomials. Discret. Math. 311, 1349–1364 (2011).
17.
Zurück zum Zitat Chen Y.Q., Xiang Q., Sehgal S.K.: An exponent bound on skew Hadamard abelian difference sets. Des. Codes Cryptogr. 4, 313–317 (1994). Chen Y.Q., Xiang Q., Sehgal S.K.: An exponent bound on skew Hadamard abelian difference sets. Des. Codes Cryptogr. 4, 313–317 (1994).
18.
Zurück zum Zitat Coulter R., Kosick P.: Commutative semifields of order 243 and 3125. Finite Fields Theory appl. Contemp. Math. 518, 129–136 (2010). Coulter R., Kosick P.: Commutative semifields of order 243 and 3125. Finite Fields Theory appl. Contemp. Math. 518, 129–136 (2010).
19.
Zurück zum Zitat Davis J.A.: Partial difference sets in p-groups. Arch. Math. 63, 103–110 (1994). Davis J.A.: Partial difference sets in p-groups. Arch. Math. 63, 103–110 (1994).
20.
Zurück zum Zitat Dillon J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974). Dillon J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974).
21.
Zurück zum Zitat Dillon J.F.: Elementary Hadamard difference sets. In: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (1975), pp. 237–249. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Manitoba (1975). Dillon J.F.: Elementary Hadamard difference sets. In: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (1975), pp. 237–249. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Manitoba (1975).
22.
Zurück zum Zitat Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Croptogr. 17, 225–235 (1999). Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Croptogr. 17, 225–235 (1999).
23.
Zurück zum Zitat Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree–Tits slice sympletic spreads in PG \((3,3^{2h+1})\). J. Comb. Theory Ser. A 114, 867–887 (2007). Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree–Tits slice sympletic spreads in PG \((3,3^{2h+1})\). J. Comb. Theory Ser. A 114, 867–887 (2007).
24.
Zurück zum Zitat Ding C., Yin J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006). Ding C., Yin J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006).
25.
Zurück zum Zitat Feng T.: Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism. J. Comb. Theory Ser. A 118, 27–36 (2011). Feng T.: Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism. J. Comb. Theory Ser. A 118, 27–36 (2011).
26.
Zurück zum Zitat Feng T., Momihara K., Xiang Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1206.3354. Feng T., Momihara K., Xiang Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1206.3354.
27.
Zurück zum Zitat Feng T., Xiang Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Comb. Theory Ser. A 119, 245–256 (2012). Feng T., Xiang Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Comb. Theory Ser. A 119, 245–256 (2012).
28.
Zurück zum Zitat Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962). Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962).
29.
Zurück zum Zitat Johnson E.C.: Skew-Hadamard abelian group difference sets. J. Algebra 4, 388–402 (1966). Johnson E.C.: Skew-Hadamard abelian group difference sets. J. Algebra 4, 388–402 (1966).
30.
Zurück zum Zitat Kantor W.M.: 2-Transitive symmetric designs. Trans. Am. Math. Soc. 146, 1–28 (1969). Kantor W.M.: 2-Transitive symmetric designs. Trans. Am. Math. Soc. 146, 1–28 (1969).
31.
Zurück zum Zitat Langevin P.: Calcus de certaines sommes de Gauss. J. Number Theory 63, 59–64 (1997). Langevin P.: Calcus de certaines sommes de Gauss. J. Number Theory 63, 59–64 (1997).
32.
Zurück zum Zitat Leung K.H., Ma S.L., Schmidt B.: Constructions of relative difference sets with classical parameters and circulant weighing matrices. J. Comb. Theory Ser. A 99, 111–127 (2002). Leung K.H., Ma S.L., Schmidt B.: Constructions of relative difference sets with classical parameters and circulant weighing matrices. J. Comb. Theory Ser. A 99, 111–127 (2002).
33.
Zurück zum Zitat Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988). Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988).
34.
Zurück zum Zitat Ma S.L.: Partial difference sets. Discret. Math. 52, 75–89 (1984). Ma S.L.: Partial difference sets. Discret. Math. 52, 75–89 (1984).
35.
Zurück zum Zitat Ma S.L.: Polynomial addition sets and symmetric difference sets. In: Ray-Chandhuri, D. (ed.) Coding Theory and Design Theory Part II: Design Theory, pp. 273–279. Springer, New York (1990). Ma S.L.: Polynomial addition sets and symmetric difference sets. In: Ray-Chandhuri, D. (ed.) Coding Theory and Design Theory Part II: Design Theory, pp. 273–279. Springer, New York (1990).
36.
Zurück zum Zitat Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994). Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994).
37.
Zurück zum Zitat Ma S.L.: Reversible relative difference sets. Combinatorica 12, 425–432 (1992). Ma S.L.: Reversible relative difference sets. Combinatorica 12, 425–432 (1992).
38.
Zurück zum Zitat Momihara K.: Skew Hadamard difference sets from cyclotomic strongly regular graphs. arXiv:1211. 2864v1. Momihara K.: Skew Hadamard difference sets from cyclotomic strongly regular graphs. arXiv:1211. 2864v1.
39.
Zurück zum Zitat Muzychuk M.: On skew Hadamard difference sets. arXiv:1012.2089v1. Muzychuk M.: On skew Hadamard difference sets. arXiv:1012.2089v1.
40.
Zurück zum Zitat Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933). Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933).
41.
Zurück zum Zitat Peisert W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001). Peisert W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001).
42.
Zurück zum Zitat Polhill J.: Paley type partial difference sets in non \(p\)-groups. Des. Codes Cryptogr. 52, 163–169 (2009). Polhill J.: Paley type partial difference sets in non \(p\)-groups. Des. Codes Cryptogr. 52, 163–169 (2009).
43.
Zurück zum Zitat Polhill J.: Paley type partial difference sets in groups of order \(n^4\) and \(9n^4\) for any odd \(n\). J. Comb. Theory Ser. A 117, 1027–1036 (2010). Polhill J.: Paley type partial difference sets in groups of order \(n^4\) and \(9n^4\) for any odd \(n\). J. Comb. Theory Ser. A 117, 1027–1036 (2010).
44.
Zurück zum Zitat Pott A.: Finite geometry and character theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin, (1995). Pott A.: Finite geometry and character theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin, (1995).
45.
Zurück zum Zitat Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007). Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007).
46.
Zurück zum Zitat Xiang Q.: Note on Paley type partial difference sets. Groups, Difference Sets, and the Monster (Columbus, OH, 1993), pp. 239–244. Ohio State University Mathematical Research Institute Publications, Berlin (1996). Xiang Q.: Note on Paley type partial difference sets. Groups, Difference Sets, and the Monster (Columbus, OH, 1993), pp. 239–244. Ohio State University Mathematical Research Institute Publications, Berlin (1996).
47.
Zurück zum Zitat Yamamoto K.: On congruences arising from relative Gauss sum. Number Theory and Combinatorics, pp. 423–446. World Scientific, Singapore (1955). Yamamoto K.: On congruences arising from relative Gauss sum. Number Theory and Combinatorics, pp. 423–446. World Scientific, Singapore (1955).
Metadaten
Titel
Paley type sets from cyclotomic classes and Arasu–Dillon–Player difference sets
verfasst von
Yu Qing Chen
Tao Feng
Publikationsdatum
01.03.2015
Verlag
Springer US
Erschienen in
Designs, Codes and Cryptography / Ausgabe 3/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-013-9881-9

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