## The Michigan Mathematical Journal

- Michigan Math. J.
- Volume 68, Issue 1 (2019), 193-210.

### 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.

#### Article information

**Source**

Michigan Math. J., Volume 68, Issue 1 (2019), 193-210.

**Dates**

Received: 6 April 2017

Revised: 20 October 2017

First available in Project Euclid: 30 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.mmj/1548817531

**Digital Object Identifier**

doi:10.1307/mmj/1548817531

**Mathematical Reviews number (MathSciNet)**

MR3934609

**Subjects**

Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 94B05: Linear codes, general

#### Citation

Higashitani, Akihiro. Lattice Simplices of Maximal Dimension with a Given Degree. Michigan Math. J. 68 (2019), no. 1, 193--210. doi:10.1307/mmj/1548817531. https://projecteuclid.org/euclid.mmj/1548817531