On lattices over Fermat function fields
Rafael Froner Prando, Pietro Speziali
TL;DR
This work constructs a novel infinite family of function-field lattices from Fermat curves $\mathcal{F}_n$ using the set of $3n$ inflection points, yielding lattices $\Lambda_n$ of rank $3n-1$ with minimum distance $d(\Lambda_n)=\sqrt{2n}$ that beat the standard bound $d\geq \sqrt{2\gamma}$ where $\gamma=n-1$. The authors establish a fixed kissing number $K(\Lambda_n)=6$ and show the lattices are not well-rounded for $n\ge 4$, while also determining the second minimum distance and the volume; they give explicit descriptions of the underlying class-group structure $\operatorname{Cl}^0(\mathcal{P})$ and derive the volume formulas $V(\Lambda_n)$. The results provide explicit high-rank function-field lattices with superior minimum distance, offering potential applications in coding theory and cryptography and motivating further exploration of lattices from function fields with favorable rational-place configurations.
Abstract
Function field lattices are an interesting example of algebraically constructed lattices. Their minimum distance is bounded below by a function of the gonality of the underlying function field. Known explicit examples--coming mostly from elliptic and Hermitian curves--typically meet this lower bound. In this paper, we construct, for every integer $n \geqslant 4$, a new family of lattices arising from the Fermat function field $F_n$ and the set of its $3n$ total inflection points. These lattices have rank $3n-1$, and we show that their minimum distance equals $\sqrt{2n}$, thereby exceeding the classical bound $\sqrt{2γ(F_n)} = \sqrt{2(n-1)}$. We also determine their kissing number, which turns out to be independent of $n$, and analyze the structure of the second shortest vectors. Our results provide the first explicit examples of function field lattices of arbitrarily large rank whose minimum distance surpasses the expected bound, offering new geometric features of potential interest for coding-theoretic and cryptographic applications.