Maximum exponent of Boolean circulant matrices with constant number of nonzero entries in their generating vector (Q1028848)

Summary: It is well-known that the maximum exponent that an \(n\)-by-\(n\) Boolean primitive circulant matrix can attain is \(n-1\). In this paper, we find the maximum exponent attained by \(n\)-by-\(n\) Boolean primitive circulant matrices with constant number of nonzero entries in their generating vector. We also give matrices attaining such exponents. Solving this problem we also solve two equivalent problems: 1) find the maximum exponent attained by primitive Cayley digraphs on a cyclic group whose vertices have constant outdegree; 2) determine the maximum order of a basis for \({\mathbb Z}_{n}\) with fixed cardinality.