Circulants and difference sets

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.