2015/11/11 by Chowdhury, Ameera · 1 citation
#05B35 #15A03 #51E21 #94B05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1511.03623
Let Fq be a finite field of order q with characteristic p. An arc is an ordered family of at least k vectors in (Fq)k in which every subfamily of size k is a basis of (Fq)k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= q, then an arc in (Fq)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 (Fq)k and a nonnegative integer n, we construct a matrix MG\uparrow n, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix MG\uparrow n to properties of the arc G and may provide new tools in the computational classification of large arcs.