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

We consider $d$-dimensional lattice polytopes $\mathrm{\Delta }$ with $h^{\ast }$-polynomial $h_{\mathrm{\Delta }}^{\ast }=1+h_k^{\ast }{t}^k$ for $1<k<(d+1)\mathrm{/}2$ and relate them to some abelian subgroups of ${\text{SL}}_{d+1}(\text{C})$ of order $1+h_k^{\ast }=p^r$ where $p$ is a prime number. These subgroups can be investigate by means of coding theory as special linear constant weight codes in ${\text{F}}_p^{d+1}$. 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 $B_{1,\chi }^{(r)}$ associated with odd characters $\chi :{\text{F}}_q^{\ast }\to {\text{C}}^{\ast }$ where $q=p^r$. Our result implies a complete classification of all lattice polytopes whose $h^{\ast }$-polynomial is a binomial.