This paper establishes a novel upper bound-termed the arithmetic Singleton bound-on the Hamming distance of any simple-root constacyclic code over a finite field. The key technical ingredient is the notion of multiple equal-difference (MED) representations of the defining set of a simple-root polynomial, which generalizes the MED representation of a cyclotomic coset. We prove that every MED representation induces an upper bound on the minimum distance; the classical Singleton bound corresponds to the coarsest representation, while the strongest among these bounds is defined as the arithmetic Singleton bound. It is shown that the arithmetic Singleton bound is always at least as tight as the Singleton bound, and a precise criterion for it to be strictly tighter is obtained. For irreducible constacyclic codes, the bound is given explicitly by $ω+1$, where $ω$ is a constant closely related to the order of $q$ modulo the radical of the polynomial order. This work provides the first systematic translation of arithmetic structure-via MED representations-into restrictive constraints on the minimum distance, revealing that the Singleton bound may be unattainable not because of linear limitations, but due to underlying algebraic obstructions.

翻译：本文针对有限域上的任意简单根常循环码，建立了一个新颖的上界——称为算术Singleton界。该界的关键技术要素是简单根多项式定义集的多重等差（MED）表示概念，它推广了分圆陪集的MED表示。我们证明每个MED表示都会导出一个最小距离的上界；经典的Singleton界对应最粗糙的表示，而这些界中最强的被定义为算术Singleton界。研究表明，算术Singleton界始终至少与Singleton界同样紧致，并获得了该界严格更紧致的精确判据。对于不可约常循环码，该界由$ω+1$显式给出，其中$ω$是与$q$模多项式阶根式的阶密切相关的常数。这项工作首次通过MED表示，系统地将算术结构转化为对最小距离的限制性约束，揭示了Singleton界可能无法达到的原因并非线性限制，而是源于潜在的代数障碍。