Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $A=(\alpha_0, \cdots, \alpha_{k-1})\in \mathbb{F}^k_{q^n}$,
let $M_{A,I}$ denote the matrix $(\alpha_i^{q^j})$ 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 $\alpha_0,\cdots,
\alpha_{k-1}$ 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 Csajb\'ok, 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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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex