Abstract
We obtain a broadly applicable decomposition of group ring elements into a “subfield part” and a “kernel part”. Applications include the verification of Lander’s conjecture for all difference sets whose order is a power of a prime >3 and for all McFarland, Spence and Chen/Davis/Jedwab difference sets. We obtain a new general exponent bound for difference sets. We show that there is no circulant Hadamard matrix of order v with 4<v<548, 964, 900 and no Barker sequence of length l with 13 < l ≤ 1022.
Similar content being viewed by others
References
K.T. Arasu S.L. Ma (1998) ArticleTitleAbelian difference sets without self-conjugacy Des. Codes Cryptogr. 15 223–230 Occurrence Handle10.1023/A:1008323907194
K.T. Arasu S.L. Ma (2001) ArticleTitleA nonexistence result on difference sets, partial difference sets and divisible difference sets J. Stat. Planning and Inference. 95 67–73 Occurrence Handle10.1016/S0378-3758(00)00278-0
K.T. Arasu S.L. Ma (2001) ArticleTitleSome new results on circulant weighing matrices J. Alg. Combin. 14 91–101 Occurrence Handle10.1023/A:1011903510338
K.T. Arasu Q. Xiang (1995) ArticleTitleMultiplier theorems J. Comb. Des. 3 257–267
Baumert L.D. (1971). Cyclic Difference Sets. Springer Lecture Notes, Springer. 182
Beth T., Jungnickel D., Lenz H. (1999). Design Theory, 2nd Edition, Cambridge University Press
S. Eliahou M. Kervaire (1992) ArticleTitleBarker sequences and difference sets L’Enseignement Math. 38 345–382
S. Eliahou M. Kervaire B. Saffari (1990) ArticleTitleA new restriction on the length of Golay complementary sequences J. Comb. Theory Ser. A. 55 49–59 Occurrence Handle10.1016/0097-3165(90)90046-Y
M. Hall (1947) ArticleTitleCyclic projective planes Duke Math. J. 14 1079–1090 Occurrence Handle10.1215/S0012-7094-47-01482-8
Ireland K., Rosen M. (1990). A Classical Introduction to Modern Number Theory. Graduate Texts in Math. Vol 84, Springer
Jacobson N. (1985). Basic Algebra I, Second Edition, W. H. Freeman and Company
Z. Jia (2002) ArticleTitleNew necessary conditions for the existence of difference sets without self-conjugacy J. Comb. Theory Ser. A. 98 312–327 Occurrence Handle10.1006/jcta.2001.3239
Jungnickel D. (1992). Difference Sets, In J. H. Dinitz and D. R. Stinson (eds), Contemporary Design Theory: A Collection of Surveys. Wiley. 241–324
Jungnickel D., Schmidt B. (1997). Difference sets: An update, In J. W. P. Hirschfeld, et al. (eds.), Geometry, Combinatorial Designs and Related Structures, Proc. First Pythagorean Conference, Cambridge University Press (1997) pp. 89–112.
T.Y. Lam K.H. Leung (2000) ArticleTitleOn Vanishing Sums of Roots of Unity J. Algebra. 224 91–109 Occurrence Handle10.1006/jabr.1999.8089
Lander E.S. (1983). Symmetric Designs: An Algebraic Approach, London Math. Soc. Lect. Notes Vol. 75, Cambridge University Press
K.H. Leung S.L. Ma B. Schmidt (2004) ArticleTitleNonexistence of abelian difference sets: Lander’s conjecture for prime power orders Trans. Amer. Math. Soc. 356 4343–4358 Occurrence Handle10.1090/S0002-9947-03-03365-8
Lidl R., Niederreiter H. (1994). Introduction to Finite Fields and Their Applications, Cambridge University Press
S.L. Ma (1996) ArticleTitlePlanar functions, relative difference sets and character theory J. Algebra. 185 342–356 Occurrence Handle10.1006/jabr.1996.0329
McFarland R.L. (1970). On Multipliers of Abelian Difference Sets, Ph.D. Dissertation, Ohio State University
Pott A. (1995). Finite Geometry and Character Theory. Springer Lecture Notes, Springer. 1601
B. Schmidt (1999) ArticleTitleCyclotomic integers and finite geometry J. Am. Math. Soc. 12 929–952 Occurrence Handle10.1090/S0894-0347-99-00298-2
B. Schmidt, Characters and cyclotomic fields in finite geometry, Lec, Notes Math., Vol. 1797 (2002).
Shoup V. NTL: A Library for doing Number Theory. http://www.shoup.net/ntl/
J. Storer R. Turyn (1961) ArticleTitleOn binary sequences, Proc Amer. Math. Soc. 12 394–399
R.J. Turyn (1965) ArticleTitleCharacter sums and difference sets Pacific J. Math. 15 319–346
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Leung, K.H., Schmidt, B. The Field Descent Method. Des Codes Crypt 36, 171–188 (2005). https://doi.org/10.1007/s10623-004-1703-7
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10623-004-1703-7