New upper bounds on the number of non-zero weights of constacyclic codes
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| Format: | Preprint |
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2024
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| _version_ | 1866914778669121536 |
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| author | Chen, Li Fu, Yuqing Liu, Hongwei |
| author_facet | Chen, Li Fu, Yuqing Liu, Hongwei |
| contents | 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]}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New upper bounds on the number of non-zero weights of constacyclic codes Chen, Li Fu, Yuqing Liu, Hongwei Information Theory 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]}. |
| title | New upper bounds on the number of non-zero weights of constacyclic codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2405.00309 |