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.

翻译：函数域格是代数构造格的一个有趣范例。其最小距离的下界由底层函数域的亏格数函数所限定。已知的显式例子——主要来源于椭圆曲线和埃尔米特曲线——通常达到该下界。本文中，我们针对每个整数 $n \geqslant 4$，构造了一个新的格族，其源于费马函数域 $F_n$ 及其全部 $3n$ 个拐点集合。这些格的秩为 $3n-1$，我们证明其最小距离等于 $\sqrt{2n}$，从而超越了经典下界 $\sqrt{2γ(F_n)} = \sqrt{2(n-1)}$。我们还确定了其接触数（该数与 $n$ 无关），并分析了次短向量的结构。我们的研究首次提供了任意大秩函数域格的显式实例，其最小距离超出预期界，为编码理论和密码学应用提供了具有潜在价值的新几何特征。