Abstract:
Define the action of $S_n \times S_n$ on the set of square matrices of order $n$ over $\mathbb{F}_{2^r}$ as follows: $A^{(g_1,g_2)} = \left(a_{g_1(i), g_2(j)}\right)$ for $A \in \left(\mathbb{F}_{2^r}\right)_{n,n}$ and $(g_1,g_2) \in S_n \times S_n$. The automorphism group $\mathrm{Aut}(A)$ of a matrix $A$ is the set of all $(g_1,g_2) \in S_n \times S_n$ such that ${A^{(g_1,g_2)} = A}$. This paper provides a description of the automorphism groups of Maximum Distance Separable (MDS) matrices. It is shown that $\mathrm{Aut}(A) = \{(g,\sigma(g)): g\in G\}$, where $G < S_n$ and $\sigma$ is a conjugation in $S_n$. If $G$ is a transitive group, then either $G$ is a regular group or $n$ is even and $G$ is a Frobenius group, which is a semidirect product of a regular Abelian group and its automorphism of order two. If $G$ is intransitive, its restrictions to orbits are isomorphic and are either regular groups or Frobenius group. Research demonstrates that the frequency of MDS matrices rises significantly in matrix classes characterized by larger automorphism groups. To improve implementation efficiency, it is preferable to select matrices with fewer distinct elements and a larger number of ones. Within classes of matrices that have nontrivial automorphism groups, MDS matrices of order $8$ over $\mathbb{F}_{2^8}$ are constructed. These include matrices with $24$ ones and $6$ distinct elements, as well as matrices with $15$ ones and $5$ distinct elements. These results strengthen the findings of P. Judon and S. Vaudenay and are analogous to those of K. Gupta, S. Pandey, and I. Ray.
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