Essential covers of the cube by hyperplanes

Linial, Nathan ; Radhakrishnan, Jaikumar (2005) Essential covers of the cube by hyperplanes Journal of Combinatorial Theory, Series A, 109 (2). pp. 331-338. ISSN 0097-3165

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A 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 that ̅η+1+1)≤e(n)≤⌈n/2⌉+1.

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Deposited On:29 Aug 2011 08:38
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