Lattice simplices of maximal dimension with a given degree

TL;DR

This paper characterizes lattice simplices of maximal dimension for a given degree, linking them to binary codes and providing counterexamples to the Cayley conjecture, while refining existing bounds.

Contribution

It provides a complete classification of simplices attaining equality in Nill's bound, connecting them to binary simplex codes and refining the inequality with a sharper bound.

Findings

Simplices of maximal dimension relate to binary simplex codes.

Such simplices serve as counterexamples to the Cayley conjecture.

A new sharper bound for the dimension-degree relation is established.

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.