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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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0691306-X
Article copyright: © Copyright 1983 American Mathematical Society