On circulant involutory and orthogonal MDS matrices over finite commutative rings

Let \(k>1\) be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), proved the non existence of \(2^k \times 2^k\) orthogonal circulant MDS matrices and involutory circulant MDS matrices over finite fields of characteristic 2. The main aim of this paper is to prove the non-existence of orthogonal circulant MDS matrices of order \(2^k\times 2^k\) and involutory circulant MDS matrices of order k over finite commutative rings of characteristic 2. Precisely, we prove that any circulant orthogonal matrix of order \(2^k\) over finite commutative rings of characteristic 2 with identity is not a MDS matrix. Moreover, some related results are also discussed. Finally, we provide some examples to prove that the assumed restrictions on our main results are not superfluous.