{"docId":9524,"paperId":9524,"url":"https:\/\/cm.episciences.org\/9524","doi":"10.2478\/cm-2021-0005","journalName":"Communications in Mathematics","issn":"1804-1388","eissn":"2336-1298","volume":[{"vid":647,"name":"Volume 29 (2021), Issue 1 (Special Issue: Ostrava Mathematical Seminar)"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-03665011","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-03665011v1","dateSubmitted":"2022-05-11 15:42:06","dateAccepted":null,"datePublished":"2021-04-30 00:00:00","titles":{"en":"Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1"},"authors":["Contreras, Daniel Uzc\u00e1tegui","Goyeneche, Dardo","Turek, Ond\u0159ej","V\u00e1clav\u00edkov\u00e1, Zuzana"],"abstracts":{"en":"It is known that a real symmetric circulant matrix with diagonal entries d \u2265 0, off-diagonal entries \u00b11 and orthogonal rows exists only of order 2d + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d \u2265 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2d + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings \u2124 m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory."},"keywords":[["General Mathematics"],"[MATH]Mathematics [math]"]}