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⊆{1,…,k}×{1,…,m}P⊆{1,…,k}×{1,…,m} and an integer δ≤min⁡{k,m}δ≤min⁡{k,m}. Let C(m,k,P,K)C(m,k,P,K) be the vector space of all k×mk×m matrices with entries ai,j=0ai,j=0 if (i,j)∉P(i,j)∉P. Let α(δ,K)α(δ,K) be the maximal dimension of a linear subspace V⊆C(m,k,P,K)V⊆C(m,k,P,K) such that all A∈V∖{0}A∈V∖{0} have rank ≥δ  . We show that known lower bounds for α(δ,K)α(δ,K), for K one (resp. several, resp. almost all) finite field give the same lower bounds for some (resp. many, resp. all) number fields.