Abstract
We show that difference sets satisfying the condition n|λ have a product property which can be exploited to construct more difference sets. Many of the newly discovered difference sets arise in precisely this way.
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Arasu KT, Chen YQ and Pott A (2006). On abelian (22m+1(2m-1 + 1), 2m(2m + 1), 2m)-difference sets. J Combin Theory Ser A 23: 1120–1137
Arasu KT, Dillon JF, Player KJ Character sum factorizations yield perfect sequences, (in preparation)
Arasu KT and Player KJ (2003). A new family of cyclic difference sets with Singer parameters in characteristic three. Des Codes Cryptogr 28: 75–91
Key JD and Assmus EF Jr (1992). Designs and their codes, Cambridge Tracts in Mathematics, vol 103. Cambridge University Press, Cambridge
Beth T, Jungnickel D and Lenz H (1999). Design theory, vol 1, 2nd edn. Cambridge University Press, Cambridge
Dillon JF (1999). Multiplicative difference sets via additive characters. Des Codes Cryptogr 17: 225–235
Dillon JF and Dobbertin H (2004). New cyclic difference sets with Singer parameters. Finite Fields Appl 10: 342–389
Davis JA and Jedwab J (1997). A unifying construction for difference sets. J Combin Theory Ser A 80: 13–78
Evans R, Hollmann HDL, Krattenthaler C and Xiang Q (1999). Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets. J Combin Theory Ser A 87: 74–119
Glynn DG (1983) Two new sequences of ovals in finite Desarguesian planes of even order. Combin Math X: 217–229. Lecture Notes in Mathematics, vol 1036, Springer-Verlag, Heidelberg
Hall M Jr (1965). A survey of difference sets. Proc Amer Math Soc 7: 975–986
Hertel D (2006) Extended Hadamard equivalence. In: Gong G et al (eds) Sequences and their applications SETA06. Lecture Notes in Computer Science, vol 4086, Springer-Verlag, pp 119–128
Jungnickel D, Schmidt B (1997) Difference sets: an update. In: Hirschfeld JWP, Magliveras SS, de Resmini MJ (eds) Geometry, combinatorial designs and related structures. Cambridge Univ. Press, pp 89–112
Ma SL (1992). Reversible relative difference sets. Combinatorica 12: 425–432
No J-S, Chung H and Yun M-S (1998). Binary pseudorandom sequences of period 2m−1 with ideal autocorrelation generated by the polynomial z d + (z + 1)d. IEEE Trans Inform Theory 44: 1278–1282
No J-S, Golomb SW, Gong G, Lee H-K and Gaal P (1998). Binary pseudorandom sequences of period 2m−1 with ideal autocorrelation. IEEE Trans Inform Theory 44: 814–817
Schmidt B (1998). Nonexistence results on Chen and Davis-Jedwab difference sets. J Algebra 202: 404–413
Washington LC (1997). Introduction to cyclotomic fields, 2nd edn. Springer-Verlag, New York
Yamamoto K (1955) On congruences arising from relative Gauss sums. In: Number theory and combinatorics. World Scientific, Singapore, pp 423–446
Yu NY, Gong G (2006) Realizations from decimation Hadamard transform for special classes of binary sequences with two-level autocorrelation. In: Helleseth T et al (eds) Coding and cryptography. Lecture Notes in Computer Science, vol 3969. Springer-Verlag, pp 371–385
Yu NY, Gong G (2006) Crosscorrelation properties of binary sequences with ideal two-level autocorrelation. In: Gong G et al (eds) Sequences and their applications SETA06. Lecture Notes in Computer Science, vol 4086. Springer-Verlag, pp 104–118
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Dedicated to Dan Hughes on the occasion of his 80th birthday.
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Arasu, K.T., Chen, Y.Q., Dillon, J.F. et al. Abelian difference sets of order n dividing λ. Des. Codes Cryptogr. 44, 307–319 (2007). https://doi.org/10.1007/s10623-007-9102-5
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DOI: https://doi.org/10.1007/s10623-007-9102-5