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

TL;DR

This paper introduces the arithmetic Singleton bound for simple-root constacyclic codes over finite fields, providing a tighter upper limit on their Hamming distance based on MED representations, and reveals algebraic obstructions to attaining the classical Singleton bound.

Contribution

It defines the arithmetic Singleton bound using MED representations, generalizes the classical bound, and offers explicit criteria for when it is tighter, advancing understanding of code distance limitations.

Findings

The arithmetic Singleton bound is at least as tight as the classical Singleton bound.

Explicit formula for irreducible codes: $ ext{distance} \,\leq\, \omega+1$.

Algebraic obstructions can prevent codes from reaching the Singleton bound.

Abstract

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 $\omega+1$, where $\omega$ 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.