Inclusion Matrices and the MDS Conjecture

Let Fq be a finite field of order q with characteristic p. An arc in Fq is an ordered family of at least k vectors in which every subfamily of size k is a basis of Fq . The MDS conjecture, which was posed by Segre in 1955, states that if k 6 q, then an arc in Fq 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 6 p, and if q is not prime, for k 6 2p− 2. To accomplish this, given an arc G ⊂ Fq and a nonnegative integer n, we construct a matrix M↑n G , which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix M↑n G to properties of the arc G and may provide new tools in the computational classification of large arcs.