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New upper bounds on the number of non-zero weights of constacyclic codes

Li Chen, Yuqing Fu, Hongwei Liu

TL;DR

This work addresses the problem of bounding the number of non-zero Hamming weights in simple-root $\lambda$-constacyclic codes over $\mathbb{F}_q$ by exploiting a large automorphism group $\langle \mu_q, \rho, \sigma_\xi \rangle$ and counting orbits on $\mathcal{C}^*$. Using Burnside's lemma, it derives explicit upper bounds for irreducible and general constacyclic codes, with a necessary-and-sufficient condition for attaining the bound, and demonstrates tightness and improvements over prior results. The authors also present two special-code classes where even tighter bounds are obtained by replacing the subgroup with larger automorphism groups, providing new avenues to construct few-weight codes. Overall, the results extend and refine earlier bounds (notably Zhang and Cao 2024 and Chen et al. 2024) and offer practical tools for designing constacyclic codes with few non-zero weights, with potential applications in secret sharing and combinatorial designs.

Abstract

For any simple-root constacyclic code $\mathcal{C}$ over a finite field $\mathbb{F}_q$, as far as we know, the group $\mathcal{G}$ generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group ${\rm Aut}(\mathcal{C})$ of $\mathcal{C}$. In this paper, by calculating the number of $\mathcal{G}$-orbits of $\mathcal{C}\backslash\{\bf 0\}$, we give an explicit upper bound on the number of non-zero weights of $\mathcal{C}$ and present a necessary and sufficient condition for $\mathcal{C}$ to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in [Zhang and Cao, FFA, 2024]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code $\mathcal{C}$ belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of $\mathcal{C}$ by substituting $\mathcal{G}$ with a larger subgroup of ${\rm Aut}(\mathcal{C})$. The results derived in this paper generalize the main results in [Chen, Fu and Liu, IEEE-TIT, 2024]}.

New upper bounds on the number of non-zero weights of constacyclic codes

TL;DR

This work addresses the problem of bounding the number of non-zero Hamming weights in simple-root -constacyclic codes over by exploiting a large automorphism group and counting orbits on . Using Burnside's lemma, it derives explicit upper bounds for irreducible and general constacyclic codes, with a necessary-and-sufficient condition for attaining the bound, and demonstrates tightness and improvements over prior results. The authors also present two special-code classes where even tighter bounds are obtained by replacing the subgroup with larger automorphism groups, providing new avenues to construct few-weight codes. Overall, the results extend and refine earlier bounds (notably Zhang and Cao 2024 and Chen et al. 2024) and offer practical tools for designing constacyclic codes with few non-zero weights, with potential applications in secret sharing and combinatorial designs.

Abstract

For any simple-root constacyclic code over a finite field , as far as we know, the group generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group of . In this paper, by calculating the number of -orbits of , we give an explicit upper bound on the number of non-zero weights of and present a necessary and sufficient condition for to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in [Zhang and Cao, FFA, 2024]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of by substituting with a larger subgroup of . The results derived in this paper generalize the main results in [Chen, Fu and Liu, IEEE-TIT, 2024]}.
Paper Structure (12 sections, 17 theorems, 98 equations)

This paper contains 12 sections, 17 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group action on a linear code
  4. $\lambda$-constacyclic codes and primitive idempotents
  5. Subgroups of the automorphism group of $\mathcal{R}^{(q)}_{n,\lambda}$
  6. Improved upper bounds
  7. An improved upper bound on the number of non-zero weights of an irreducible constacyclic code
  8. An improved upper bound on the number of non-zero weights of a general constacyclic code
  9. New upper bounds on the number of non-zero weights of two special classes of $\lambda$-constacyclic codes
  10. New upper bound on the number of non-zero weights of the negacyclic code $\mathcal{C}=\mathcal{R}^{(q)}_{n,\lambda}\varepsilon_{t}\bigoplus \mu_{-1}(\mathcal{R}^{(q)}_{n,\lambda}\varepsilon_{t})$
  11. New upper bound on the number of non-zero weights of the constacyclic code $\mathcal{C}_{l_{0}}=\mathcal{R}^{(q)}_{n,\lambda}\varepsilon_{t}\bigoplus \mu_{(-1)^{l_{0}}p^{\frac{e}{2}}}(\mathcal{R}^{(q)}_{n,\lambda}\varepsilon_{t})$
  12. Concluding remarks and future works

Key Result

Lemma 2.1

(7) Let $\mathcal{C}$ be a linear code of length $n$ over $\mathbb{F}_{q}$ with $\ell$ non-zero weights and let ${\rm Aut}(\mathcal{C})$ be the automorphism group of $\mathcal{C}$. Suppose that $\mathcal{G}$ is a subgroup of ${\rm Aut}(\mathcal{C})$. If the number of $\mathcal{G}$-orbits of $\mathca

Theorems & Definitions (43)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Corollary 3.4
  • proof
  • ...and 33 more