Abstract
By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q 3k−1)/(q−1), q−1, q 3k−1, q 3k−2) relative difference sets, where q=3e. These relative difference sets are “liftings” of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K.T. Arasu, H.D. L.Hollmann, K. Player and Q.Xiang, Codes and Designs (Columbus, OH, 2000), 9–35, Ohio State Univ. Math. Res. Inst. Publ. 10, de Gruyter, Berlin (2002).
B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Wiley Interscience (1998).
T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. 1, Second edition, Cambridge University Press, Cambridge (1999).
R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., Vol. 6 (1942) pp. 1–15.
J. Cannon and C. Playoust, An Introduction to MAGMA, University of Sydney, Australia (1993).
J. E. H. Elliott and A. T. Butson, Relative difference sets, Illinois J. Math., Vol. 10 (1966) pp. 517–531.
R. Evans, H. D. L. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums and p-ranks of difference sets, J. Combin. Theory Ser. A, Vol. 87 (1999) pp. 74–119.
J. M. Goethals and P. Delsarte, On a class of majority logic decodable cyclic codes, IEEE Trans. Inform. Theory, Vol. 14 (1968) pp. 182–188.
B. Gordon, W.H. Mills and L.R. Welch, Some new difference sets, Canad. J. Math., Vol. 14 (1962) pp. 614–625.
T. Helleseth, P.V. Kumar and H. M. Martinsen, A new family of ternary sequences with ideal twolevel autocorrelation, Designs, Codes and Cryptography, Vol. 23, No. 2 (2001) pp. 157–166.
D. Jungnickel and V. D. Tonchev, Decompositions of difference sets, J. Algebra, Vol. 217 (1999) pp. 21–39.
C. W. H. Lam, On relative difference sets, In Proc. 7th Manitoba Conference on Numerical Mathematics andComputing (1977) pp. 445–474.
S. Lang, Cyclotomic Fields, Springer-Verlag, New York (1978).
J. MacWilliams and H. B. Mann, On the p-rank of the design matrix of a difference set, Inform. Control, Vol. 12 (1968) pp. 474–488.
J.-S. No, New cyclic difference sets with Singer parameters constructed from d-homogeneous functions, preprint.
J.-S. No, D.-J. Shin and T. Helleseth, On the p-ranks andcharacteristic polynomials of cyclic difference sets, preprint.
A. Pott, Finite geometry andcharacter theory, Springer LNM 1601 (1995).
J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. AMS, Vol. 43 (1938) pp. 377–385.
K. J. C. Smith, On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry, J. Combin. Theory, Vol. 7 (1969) pp. 122–129.
E. Spence, Hadamard matrices from relative difference sets, J. Combin. Theory, Vol. 19 (1975) pp. 287–300.
M. Yamada, On a relation between a cyclic relative difference sets associated with the quadratic extensions of a finite field and the Szekeres difference set, Combinatorica, Vol. 8 (1988) pp. 207–216.
K. Yamamoto, On congruences arising from relative Gauss sums, In Number Theory and Combinatorics, Japan 1984, World Scientific Publ. (1985) pp. 423–446.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chandler, D.B., Xiang, Q. Cyclic Relative Difference Sets and their p-Ranks. Designs, Codes and Cryptography 30, 325–343 (2003). https://doi.org/10.1023/A:1025750228679
Issue Date:
DOI: https://doi.org/10.1023/A:1025750228679