Essential covers of the cube by hyperplanes

AbstractA set L of linear polynomials in variables X1,X2,…,Xn with real coefficients is said to be an essential cover of the cube {0,1}n if(E1)for each v∈{0,1}n, there is a p∈L such that p(v)=0;(E2)no proper subset of L satisfies (E1), that is, for every p∈L, there is a v∈{0,1}n such that p alone takes the value 0 on v;(E3)every variable appears (in some monomial with non-zero coefficient) in some polynomial of L.Let e(n) be the size of the smallest essential cover of {0,1}n. In the present note we show that12(4n+1+1)⩽e(n)⩽n2+1