New quantum codes from homothetic-BCH codes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912823654744064 |
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| author | Galindo, Carlos Hernando, Fernando Martín-Cruz, Helena |
| author_facet | Galindo, Carlos Hernando, Fernando Martín-Cruz, Helena |
| contents | We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $\mathcal{C}$ of length $λn_1$, where $λ$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $λn_1 - n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $\mathcal{B}$ is Hermitian self-orthogonal, so is $\mathcal{C}$.
As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table.
To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New quantum codes from homothetic-BCH codes Galindo, Carlos Hernando, Fernando Martín-Cruz, Helena Information Theory We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $\mathcal{C}$ of length $λn_1$, where $λ$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $λn_1 - n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $\mathcal{B}$ is Hermitian self-orthogonal, so is $\mathcal{C}$. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table. To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths. |
| title | New quantum codes from homothetic-BCH codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2503.13069 |