Circulants and difference sets

Abstract

Let F F be any field, f ( x ) f(x) a polynomial over F F of degree ⩽ υ − 1 \leqslant \upsilon - 1 , P P the υ × υ \upsilon \times \upsilon full cycle, and C C the υ × υ \upsilon \times \upsilon circulant f ( P ) f(P) . Assume that if F F is of finite characteristic p p . then ( p , υ ) = 1 (p,\upsilon ) = 1 . It is shown that the rank of C C over F F is υ − d \upsilon - d , where d d is the degree of the greatest common divisor of f ( x ) f(x) and x υ − 1 {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.