Asymptotics of Moore exponent sets (Q778719)

Consider a general Moore matrix \(M\), that is a \(k\times k\) matrix \[ M(\alpha):=\left( \begin{matrix} \alpha_0 & \alpha_0^{q} & \cdots & \alpha_0^{q^{k-1}} \\ \alpha_1 & \alpha_1^{q} & \cdots & \alpha_1^{q^{k-1}} \\ \vdots& \vdots & \ddots & \vdots \\ \alpha_{k-1} & \alpha_{k-1}^{q} & \cdots & \alpha_{k-1}^{q^{k-1}} \end{matrix} \right) \] for a given vector \(\alpha=(\alpha_0,\dots,\alpha_{k-1})\in\mathbb{F}_{q^n}^k\) , i.e. \(k\times k\) square matrix whose \(i\)-th column is given by the \(q^{i-1}\)-th power of \(\alpha\). A very famous result regarding Moore matrices states that \(\det(M(\alpha))=0\) if and only if \(\alpha_0,\dots,\alpha_{k-1}\) are \(\mathbb{F}_q\) linearly dependent. In this paper, the authors investigate a generalization of this result to matrices having shape, for a given \(I=(i_0,\dots,i_{k-1})\in\mathbb{N}^k\) \[ M_{\alpha,I}:=\left( \begin{matrix} \alpha_0^{q^{i_0}} & \alpha_0^{q^{i_1}} & \cdots & \alpha_0^{q^{i_{k-1}}} \\ \alpha_1^{q^{i_0}} & \alpha_1^{q^{i_1}} & \cdots & \alpha_1^{q^{i_{k-1}}} \\ \vdots& \vdots & \ddots & \vdots \\ \alpha_{k-1}^{q^{i_0}} & \alpha_{k-1}^{q^{i_1}} & \cdots & \alpha_{k-1}^{q^{i_{k-1}}} \end{matrix} \right). \] The authors provide \(k\)-tuples \(I=\{0,i_1,\dots,i_{k-1}\}\) (known as Moore exponent sets) such that if \(\det(M(\alpha,i))=0\) then \(\alpha_0,\dots,\alpha_{k-1}\) are \(\mathbb{F}_q\) linearly dependent. This results has non-trivial implications in rank-metric codes, since it is strictly bounded to MRD-codes (i.e. the \(q\)-analog of MDS codes).