Vol. 5, No. 2, 2012

 Recent Issues
 The Journal Cover Page Editorial Board Editors’ Addresses Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Ethics Statement Subscriptions Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)
Distribution of the exponents of primitive circulant matrices in the first four boxes of $\mathbb{Z}_n$

Maria Isabel Bueno, Kuan-Ying Fang, Samantha Fuller and Susana Furtado

Vol. 5 (2012), No. 2, 187–205
Abstract

We consider the problem of describing the possible exponents of $n$-by-$n$ boolean primitive circulant matrices. It is well known that this set is a subset of $\left[1,n-1\right]$ and not all integers in $\left[1,n-1\right]$ are attainable exponents. In the literature, some attention has been paid to the gaps in the set of exponents. The first three gaps have been proven, that is, the integers in the intervals $\left[\frac{n}{2}+1,n-2\right]$, $\left[\frac{n}{3}+2,\frac{n}{2}-2\right]$ and $\left[\frac{n}{4}+3,\frac{n}{3}-2\right]$ are not attainable exponents. Here we study the distribution of exponents in between those gaps by giving the exact exponents attained there by primitive circulant matrices. We also study the distribution of exponents in between the third gap and our conjectured fourth gap. It is interesting to point out that the exponents attained in between the ($i-1$)-th and the $i$-th gap depend on the value of $n\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}i$.

Keywords
exponent, primitive circulant matrix, basis of a cyclic group, order, box
Mathematical Subject Classification 2010
Primary: 05C25, 05C50, 11P70