Abstract. We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2 , where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n 2 .For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes.We further give a characterisation of the roots of the Ehrhart polyomials in the 3-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.
A set of Boolean functions is called a bent set if the sum of any two distinct members is a bent function. We show that any bent set yields a homogeneous system of linked symmetric designs with the same design parameters as those systems derived from Kerdock sets. Further we observe that there are bent sets of size equal to the square root of the Kerdock set size which consist of Boolean functions with arbitrary degrees.
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