Inclusion Matrices and the MDS Conjecture

Let $\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \leq q$, then an arc in $\mathbb{F}_{q}^{k}$ 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 $k \leq p$, and if $q$ is not prime, for $k \leq 2p-2$. To accomplish this, given an arc $G \subset \mathbb{F}_{q}^{k}$ 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.