University of Isfahan Transactions on Combinatorics 2251-8657 15 4 2025 11 20 Circulant matrices of each rank over finite fields 235 250 30045 10.22108/toc.2025.138779.2097 EN Yangjiang Wei School of Mathematics and Statistics, Nanning Normal University, and Center for Applied Mathematics of Guangxi at Nanning Normal University, Nanning, P. R. China Yi Ming Zou Department of Mathematical Sciences, University of Wisconsin, Milwaukee, USA Journal Article 2023 08 15 We consider the enumeration of circulant matrices over a finite field with $q$ elements, and we provide formulas to compute the number of these circulant matrices for each rank. We also consider the computation of the orders of $q$ in multiplicative groups of integers modulo $n$ that are needed in the enumeration, and we give a reduction method to compute these orders. We then use these results to derive explicit formulas for the enumerations of infinite sequences of circulant matrices of some special types. https://toc.ui.ac.ir/article_30045_edde931952a67fae8d88621f2be8dee1.pdf