Summary: In V. Cauchois and P. Loidreau [Des. Codes Cryptography 87, No. 2–3, 249–260 (2019; Zbl 1454.94122)] gave a necessary and sufficient condition for the existence of circulant MDS matrices over a field of given characteristic. In this paper, we prove the non-existence of certain orthogonal circulant MDS matrices. Then we give a necessary and sufficient condition for orthogonal \(\theta\)-circulant matrices using \(q\)-polynomial rings. We also discuss orthogonal circulant MDS matrices over Galois rings.