Asymptotics of Moore exponent sets

Let $n$ be a positive integer and $I$ a $k$-subset of integers in $[0,n-1]$. Given a $k$-tuple $A=(α_0, \cdots, α_{k-1})\in \mathbb{F}^k_{q^n}$, let $M_{A,I}$ denote the matrix $(α_i^{q^j})$ with $0\leq i\leq k-1$ and $j\in I$. When $I=\{0,1,\cdots, k-1\}$, $M_{A,I}$ is called a Moore matrix which was introduced by E. H. Moore in 1896. It is well known that the determinant of a Moore matrix equals $0$ if and only if $α_0,\cdots, α_{k-1}$ are $\mathbb{F}_q$-linearly dependent. We call $I$ that satisfies this property a Moore exponent set. In fact, Moore exponent sets are equivalent to maximum rank-distance (MRD) code with maximum left and right idealisers over finite fields. It is already known that $I=\{0,\cdots, k-1\}$ is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered by Csajbók, Marino, Polverino and the second author both give rise to new Moore exponent sets. By using algebraic geometry approach, we obtain an asymptotic classification result: for $q>5$, if $I$ is not an arithmetic progression, then there exist an integer $N$ depending on $I$ such that $I$ is not a Moore exponent set provided that $n>N$.

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