Lattice simplices of maximal dimension with a given degree

Abstract

It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension $(4s-2)$ with degree $s$ which are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as "Cayley conjecture". Moreover, by modifying Nill's inequaitly slightly, we also see the sharper bound $d+1 \leq f(2s)$, where $f(M)=\sum_{n=0}^{\lfloor \log_2 M \rfloor} \lfloor M/2^n \rfloor$ for $M \in \mathbb{Z}_{\geq 0}$. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.