Asymptotics of Moore exponent sets

TL;DR

This paper studies the asymptotic behavior of Moore exponent sets, which are linked to maximum rank-distance codes, showing that non-arithmetic sets cease to be Moore exponent sets as the field size grows.

Contribution

It provides an algebraic geometry-based asymptotic classification of Moore exponent sets, demonstrating that non-arithmetic sets are not Moore exponent sets for sufficiently large field extensions.

Findings

Non-arithmetic Moore exponent sets do not exist beyond a certain size for q>5.

The classification uses algebraic geometry techniques.

Arithmetic progression sets remain Moore exponent sets asymptotically.

Abstract

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 $n>N$.