Construction of all MDS and involutory MDS matrices

In this paper, we propose two algorithms for a hybrid construction of all \(n\times n\) MDS and involutory MDS matrices over a finite field \(\mathbb{F}_{p^m}\), respectively. The proposed algorithms effectively narrow down the search space to identify \((n-1) \times (n-1)\) MDS matrices, facilitating the generation of all \(n \times n\) MDS and involutory MDS matrices over \(\mathbb{F}_{p^m}\). To the best of our knowledge, existing literature lacks methods for generating all \(n\times n\) MDS and involutory MDS matrices over \(\mathbb{F}_{p^m}\). In our approach, we introduce a representative matrix form for generating all \(n\times n\) MDS and involutory MDS matrices over \(\mathbb{F}_{p^m}\). The determination of these representative MDS matrices involves searching through all \((n-1)\times (n-1)\) MDS matrices over \(\mathbb{F}_{p^m}\). Our contributions extend to proving that the count of all \(3\times 3\) MDS matrices over \(\mathbb{F}_{2^m}\) is precisely \((2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)\). Furthermore, we explicitly provide the count of all \(4\times 4\) MDS and involutory MDS matrices over \(\mathbb{F}_{2^m}\) for \(m=2, 3, 4\).