Inclusion Matrices and the MDS Conjecture

TL;DR

This paper investigates the MDS conjecture in finite fields by introducing inclusion matrices related to arcs, providing new proofs for certain cases and proposing a conjecture that could confirm the conjecture for most k when q is odd.

Contribution

It introduces a new matrix construction linked to arcs and relates its algebraic properties to the MDS conjecture, offering simpler proofs and a potential path to its resolution.

Findings

Proved the conjecture for k <= p and for q not prime with k <= 2p-2.

Constructed matrices related to inclusion matrices that encode arc properties.

Provided tools for computational classification of large arcs.

Abstract

Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of at least k vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= q, then an arc in (F_q)^k has size at most q+1, unless q is even and k=3 or k=q-1, in which case it has size at most q+2. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of k when q is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when k <= p, and if q is not prime, for k <= 2p-2. To accomplish this, given an arc G of (F_q)^k and a nonnegative integer n, we construct a matrix M_G^{\uparrow n}, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix M_G^{\uparrow n} to properties of the arc G and may provide new tools in the computational classification of large arcs.