Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Circulants and difference sets

Author: Morris Newman
Journal: Proc. Amer. Math. Soc. 88 (1983), 184-188
MSC: Primary 05B10; Secondary 05B20, 12C15
MathSciNet review: 691306
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F$ be any field, $ f(x)$ a polynomial over $ F$ of degree $ \leqslant \upsilon - 1$, $ P$ the $ \upsilon \times \upsilon $ full cycle, and $ C$ the $ \upsilon \times \upsilon $ circulant $ f(P)$. Assume that if $ F$ is of finite characteristic $ p$. then $ (p,\upsilon ) = 1$. It is shown that the rank of $ C$ over $ F$ is $ \upsilon - d$, where $ d$ is the degree of the greatest common divisor of $ f(x)$ and $ {x^\upsilon } - 1$. This result is used to determine the rank modulo a prime of the incidence matrix associated with a difference set. The notion of the degree of a difference set is introduced. Certain theorems connected with this notion are proved, and an open problem is stated. Some numerical results are appended.

References [Enhancements On Off] (What's this?)

  • [1] M. Hall, Jr., A survey of difference sets, Proc. Amer. Math. Soc. 7 (1956), 975-986. MR 0082502 (18:560h)
  • [2] M. Newman, Invariant factors of combinatorial matrices, Israel J. Math. 10 (1971), 126-130. MR 0369103 (51:5339)
  • [3] H. J. Ryser, Combinatorial mathematics, Carus Math. Monographs, no. 14, Math. Assoc. Amer., distributed by Wiley, New York, 1963. MR 0150048 (27:51)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 05B10, 05B20, 12C15

Retrieve articles in all journals with MSC: 05B10, 05B20, 12C15

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society