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=(α0,⋯,αk−1)∈Fqjavax.xml.bind.JAXBElement@58b36835k, let MA,I denote the matrix (αiqjavax.xml.bind.JAXBElement@58d19926) with 0≤i≤k−1 and j∈I. When I={0,1,⋯,k−1}, MA,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 α0,⋯,αk−1 are Fq-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 idealizers over finite fields. It is already known that I={0,⋯,k−1} is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered in [5] 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 exists an integer N depending on I such that I is not a Moore exponent set provided that n>N.