Cyclic relative difference sets and their p-ranks

Designs, Codes, and Cryptography

Volumn 30, Issue 3, 2003, Pages 325-343

  Chandler, David B ^a   Xiang, Qing ^a  

^a UNIVERSITY OF DELAWARE   (United States)

Author keywords

Affine GMW difference set; Gauss sum; Relative difference set; Singer difference set; Stickelberger's theorem; Teichm ller character

Indexed keywords

EQUIVALENCE CLASSES; FOURIER TRANSFORMS; MATRIX ALGEBRA; POLYNOMIALS; SET THEORY; THEOREM PROVING; VECTORS;

AFFINE DIFFERENCE SET; CYCLIC CODES; CYCLIC RELATIVE DIFFERENCE SETS; GAUSS SUM; SINGER DIFFERENCE SET; STICKELBERGER THEOREM; TEICHMULLER CHARACTER;

CODES (SYMBOLS);

EID: 0242302660     PISSN: 09251022     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1025750228679     Document Type: Article

References (22)

1. 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).
2. B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Wiley Interscience (1998).
3. T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. 1, Second edition, Cambridge Univesity Press, Cambridge (1999).
4. R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., Vol. 6 (1942) pp. 1-15.
5. J. Cannon and C. Playoust, An Introduction to MAGMA, University of Sydney, Australia (1993).
6. J. E. H. Elliott and A. T. Butson, Relative difference sets, Illinois J. Math., Vol. 10 (1966) pp. 517-531.
7. 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.
8. 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.
9. B. Gordon, W. H. Mills and L. R. Welch, Some new difference sets, Canad. J. Math., Vol. 14 (1962) pp. 614-625.
10. T. Helleseth, P. V. Kumar and H. M. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation, Designs, Codes and Cryptography, Vol. 23, No. 2 (2001) pp. 157-166.
11. D. Jungnickel and V. D. Tonchev, Decompositions of difference sets, J. Algebra, Vol. 217 (1999) pp. 21-39.
12. C. W. H. Lam, On relative difference sets, In Proc. 7th Manitoba Conference on Numerical Mathematics and Computing (1977) pp. 445-474.
13. S. Lang, Cyclotomic Fields, Springer-Verlag, New York (1978).
14. 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.
15. J.-S. No, New cyclic difference sets with Singer parameters constructed from d-homogeneous functions, preprint.
16. J.-S. No, D.-J. Shin and T. Helleseth, On the p-ranks and characteristic polynomials of cyclic difference sets, preprint.
17. A. Pott, Finite geometry and character theory, Springer LNM 1601 (1995).
18. J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. AMS, Vol. 43 (1938) pp. 377-385.
19. 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.
20. E. Spence, Hadamard matrices from relative difference sets, J. Combin. Theory, vol. 19 (1975) pp. 287-300.
21. 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.
22. K. Yamamoto, On congruences arising from relative Gauss sums, In Number Theory and Combinatorics, Japan 1984, World Scientific Publ. (1985) pp. 423-446.