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

Li Zhu, Hongfeng Wu

TL;DR

This work addresses how the arithmetic structure of simple-rooted constacyclic codes over a finite field constrains their minimum distance beyond the classical Singleton bound. It introduces multiple equal-difference (MED) representations of a code's defining set and shows that each MED representation yields an upper bound on the Hamming distance; the weakest bound recovers the Singleton bound, while the strongest is the arithmetic Singleton bound. For irreducible constacyclic codes, the arithmetic Singleton bound is explicitly $b_{AS}(\\mathcal{C})=\\omega+1$, with $\\omega$ determined by the order of $q$ modulo the radical of the order of the generator polynomial, and the classical bound is shown to match the AS bound precisely when certain valuation conditions hold. The results reveal that the Singleton bound may be unattainable not due to linear constraints but due to deeper algebraic obstructions, and they connect MED representations to binomial factorizations over extension fields, providing a systematic algebraic framework for distance constraints and potential extensions to broader code families.

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 $ω+1$, 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.

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

TL;DR

This work addresses how the arithmetic structure of simple-rooted constacyclic codes over a finite field constrains their minimum distance beyond the classical Singleton bound. It introduces multiple equal-difference (MED) representations of a code's defining set and shows that each MED representation yields an upper bound on the Hamming distance; the weakest bound recovers the Singleton bound, while the strongest is the arithmetic Singleton bound. For irreducible constacyclic codes, the arithmetic Singleton bound is explicitly , with determined by the order of modulo the radical of the order of the generator polynomial, and the classical bound is shown to match the AS bound precisely when certain valuation conditions hold. The results reveal that the Singleton bound may be unattainable not due to linear constraints but due to deeper algebraic obstructions, and they connect MED representations to binomial factorizations over extension fields, providing a systematic algebraic framework for distance constraints and potential extensions to broader code families.

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 , where is a constant closely related to the order of 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.
Paper Structure (9 sections, 24 theorems, 97 equations, 1 table)

This paper contains 9 sections, 24 theorems, 97 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The multiple equal-difference representations of the defining set of a polynomial
  4. A reminder on the MED representations of a cyclotomic coset
  5. The MED representations of the defining set of a polynomial
  6. The arithmetic Singleton bound on the Hamming distances of simple-rooted constacyclic codes over finite fields
  7. Arithmetic Singleton bound
  8. The arithmetic Singleton bound for irreducible constacyclic codes
  9. Conclusion

Key Result

Lemma 2.1

Nezami Let $\ell$ be an odd prime number, and $m$ be an integer such that $\ell \mid m-1$. Then $v_{\ell}(m^{d}-1) = v_{\ell}(m-1) + v_{\ell}(d)$ for any positive integer $d$.

Theorems & Definitions (43)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Proposition 3.6
  • ...and 33 more