The Arithmetic Singleton Bound on the Hamming Distances of Simple-rooted Constacyclic Codes over Finite Fields

Abstract

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.