In this work, We introduce a new upper bound on the Hamming distance of simple-root constacyclic codes over finite fields, which we call the arithmetic Singleton bound. The main technical tool is the notion of a multiple equal-difference (MED) representation. Via the MED representations of the defining set of the generator polynomial of a simple-root constacyclic code, we obtain a family of upper bounds on its Hamming distance, among which the weakest one coincides with the Singleton bound, while the strongest one is defined to be the arithmetic Singleton bound for this code. Consequently, the arithmetic Singleton bound is always at least as strong as the classical Singleton bound, and is in fact strictly stronger in numerous nontrivial cases. The arithmetic Singleton bound partially measures the restriction on the Hamming distance of a simple-root constacyclic code imposed by its arithmetic structure. In particular, for an irreducible constacyclic code, the MED representations of the defining set of its generator polynomial are completely determined, via which the arithmetic Singleton bound is computed concretely. Finally for any simple-root cyclic code the arithmetic Singleton bound and the BCH bound are compared.

翻译：本文提出有限域上单根常循环码汉明距离的一个新上界，称为算术Singleton界。主要技术工具是多重等差(MED)表示的概念。通过单根常循环码生成多项式定义集的MED表示，我们得到其汉明距离的一族上界，其中最弱者与Singleton界一致，而最强者被定义为该码的算术Singleton界。因此，算术Singleton界始终不弱于经典Singleton界，并且在许多非平凡情形下严格强于后者。算术Singleton界部分度量了单根常循环码算术结构对其汉明距离的限制。特别地，对于不可约常循环码，其生成多项式定义集的MED表示被完全确定，从而可具体计算算术Singleton界。最后，对任意单根循环码，比较了算术Singleton界与BCH界。