Circulant matrices of each rank over finite fields

Document Type : Research Paper

Authors

1 School of Mathematics and Statistics, Nanning Normal University, and Center for Applied Mathematics of Guangxi at Nanning Normal University, Nanning, P. R. China

2 Department of Mathematical Sciences, University of Wisconsin, Milwaukee, USA

Abstract

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.

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References

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Transactions on Combinatorics
Volume 15, Issue 4 - Serial Number 4
December 2026
Pages 235-250

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History

  • Receive Date: 15 August 2023
  • Revise Date: 06 November 2025
  • Accept Date: 15 November 2025
  • Published Online: 20 November 2025

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