On lattices over Fermat function fields

TL;DR

This paper constructs a new family of high-rank lattices from Fermat function fields that surpass known minimum distance bounds, with potential applications in coding theory and cryptography.

Contribution

It introduces explicit lattices from Fermat function fields with minimum distance exceeding classical bounds, expanding the understanding of algebraic lattice constructions.

Findings

Minimum distance equals √(2n), surpassing classical bounds

Kissing number is independent of n

Analyzes structure of second shortest vectors

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 $\sqrt{2\gamma(F_n)} = \sqrt{2(n-1)}$. 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.