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 $s$ that is not a lattice pyramid, we have $d+1\le 4s-1$. In this paper, we give a complete characterization of lattice simplices satisfying the equality, that is, the lattice simplices of dimension $4s-2$ with degree $s$ that 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 the Cayley conjecture. Moreover, by slightly modifying Nill’s inequality we also see the sharper bound $d+1\le f(2s)$, where $f(M)={\sum }_{n=0}^{\lfloor {log}_{2}M\rfloor }\lfloor M/2^n\rfloor$ for $M\in {\mathbb{Z}}_{\ge 0}$. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.