Characterizing hyperplane coverings of vector spaces over the binary field

Let V be a vector space over the binary field F2={0,1} with dimV≥2. If W={W1,…,Wn} is a family of hyperplanes of V, one says that W is an irredundant covering for V provided that W covers V, that is, ∪i=1nWi=V and no proper subfamily of W covers V. We show that W is an irredundant covering for V if and only if the following three conditions are satisfied: (i) n is odd, (ii) dimV/∩i=1nWi=n−1, and (iii) any (n−1)-fold intersection of the elements of W has codimension n – 1. Moreover, we show that for any odd integer n with 3≤n≤dimV+1 there is an irredundant hyperplane covering for V with n elements.