Lattice polytopes, finite abelian subgroups in (n,) and coding theory

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We consider d-dimensional lattice polytopes Δ with h*-polynomial h*Δ=1+hₖ*tᵏ for 1<k<(d+1)/2 and relate them to some abelian subgroups of ₊₁() of order 1+hₖ*=pʳ where p is a prime number. These subgroups can be investigate by means of coding theory as special linear constant weight codes in ₚᵈ⁺¹. If p =2, then the classication of these codes and corresponding lattice polytopes can be obtained using a theorem of Bonisoli. If p > 2, the main technical tool in the classification of these linear codes is the non-vanishing theorem for generalized Bernoulli numbers B₁,ᵪ⁽ʳ⁾ associated with odd characters χ:→ where q=pʳ. Our result implies a complete classification of all lattice polytopes whose h*-polynomial is a binomial.