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We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial$h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abeliansubgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a primenumber. These subgroups can be investigate by means of coding theory as speciallinear constant weight codes in $\F_p^{d+1}$. If $p =2$, then the classicationof these codes and corresponding lattice polytopes can be obtained using atheorem of Bonisoli. If $p > 2$, the main technical tool in the classificationof these linear codes is the non-vanishing theorem for generalized Bernoullinumbers $B_{1,\chi}^{(r)}$ associated with odd characters $\chi:\F_q^*\to\C^*$where $q=p^r$. Our result implies a complete classification of all latticepolytopes whose $h^*$-polynomial is a binomial.

Lattice polytopes, finite abelian subgroups in $\SL(n,\C)$ and coding theory