Linear subspaces of matrices associated to a Ferrers diagram and with a prescribed lower bound for their rank

Fix a field K, a subset P subset of {1,, ..,k} x {1,...,m} and an integer delta <= min{k, m}. Let C(m, k, P, K) be the vector space of all k x m matrices with entries a(i,j) = 0 if (i, j) is not an element of P. Let alpha(delta, K) be the maximal dimension of a linear subspace V subset of C(m, k, P, K) such that all A is an element of V \ {0} have rank >= delta. We show that known lower bounds for alpha(delta, K), for K one (resp. several, resp. almost all) finite field give the same lower bounds for some (resp. many, resp. all) number fields. (C) 2015 Elsevier Inc. All rights reserved.