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Some bounds on the cardinality of the $b$-symbol weight spectrum of codes

Hongwei Zhu, Shitao Li, Minjia Shi, Shu-Tao Xia, Patrick Sole

TL;DR

The paper generalizes bounds on the distance spectrum from the Hamming metric to the $b$-symbol metric across unrestricted, additive, linear, and cyclic codes. It introduces three complementary approaches—period distribution, primitive idempotents, and a $b$-symbol weight formula—for bounding the $b$-symbol weight spectrum of cyclic codes, and provides unified results connecting spectrum size to combinatorial structures like Singer difference sets and Golomb rulers. As by-products, it determines the maximum number of symplectic weights for linear codes and proves a fundamental, length- and dimension-based lower bound on the minimum distance of cyclic codes. The findings offer useful bounds and constructions that can guide code design under the $b$-symbol metric and expose connections to classical combinatorial objects. The work lays groundwork for future exploration of M$b$SW codes and extensions to reducible cyclic codes, with potential impact on robust data storage and quantum-code related constructions via the symplectic metric.

Abstract

The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the $b$-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the $b$-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the $b$-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the $b$-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters $[n,k,d_H(\C)]_q$ of cyclic codes is provided.

Some bounds on the cardinality of the $b$-symbol weight spectrum of codes

TL;DR

The paper generalizes bounds on the distance spectrum from the Hamming metric to the -symbol metric across unrestricted, additive, linear, and cyclic codes. It introduces three complementary approaches—period distribution, primitive idempotents, and a -symbol weight formula—for bounding the -symbol weight spectrum of cyclic codes, and provides unified results connecting spectrum size to combinatorial structures like Singer difference sets and Golomb rulers. As by-products, it determines the maximum number of symplectic weights for linear codes and proves a fundamental, length- and dimension-based lower bound on the minimum distance of cyclic codes. The findings offer useful bounds and constructions that can guide code design under the -symbol metric and expose connections to classical combinatorial objects. The work lays groundwork for future exploration of MSW codes and extensions to reducible cyclic codes, with potential impact on robust data storage and quantum-code related constructions via the symplectic metric.

Abstract

The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the -symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the -symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the -symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the -symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters of cyclic codes is provided.
Paper Structure (15 sections, 30 theorems, 56 equations, 1 figure)

This paper contains 15 sections, 30 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notation fixed throughout this paper
  4. The Hamming metric and the $b$-symbol metric
  5. Group action and automorphism group
  6. Some basics about periodic sequences
  7. Unrestricted codes
  8. Addtive codes
  9. Linear codes
  10. Cyclic codes
  11. The period distribution approach
  12. The primitive idempotent approach
  13. The $b$-symbol weight formula approach
  14. Performance analysis of three approaches
  15. Conclusion and future works

Key Result

Lemma 3

BUG For any vector $\mathbf{a}=(a_0,a_1,\ldots,a_{n-1})\in\mathbb{F}_q^n,$ we have

Figures (1)

  • Figure 1: The Venn diagram of these codes

Theorems & Definitions (63)

  • Definition 2
  • Lemma 3
  • Example 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Definition 7
  • Theorem 8
  • Theorem 9
  • ...and 53 more