Essential covers of the cube by hyperplanes

A set L of linear polynomials in variables X₁, X₂, ..., X_n with real coefficients is said to be an essential cover of the cube {0, 1}ⁿ if (E1) for each ν ∈ {0, 1}ⁿ, there is a p ∈ L such that p (ν) = 0; (E2) no proper subset of L satisfies (E1), that is, for every p ∈ L, there is a ν ∈ {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 that 1/2 (√4n+1 + 1) le; e(n) ≤ ⌈n/2⌉ + 1.