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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.07740 |
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| _version_ | 1866911875551199232 |
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| author | Cao, Meng |
| author_facet | Cao, Meng |
| contents | Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the $σ$ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be $σ$ dual-containing and $σ$ self-orthogonal. We prove that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))$. We prove that for any integer $h$ with $\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})$, there exists a linear code $\mathcal{C}_{2,h}$ monomially equivalent to $\mathcal{C}_{2}$ such that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=h$, where $\mathcal{C}_{i}$ is an $[n,k_{i}]_{q}$ linear code for $i=1,2$. We show that given an $[n,k,d]_{q}$ linear code $\mathcal{C}$, there exists a monomially equivalent $[n,k,d]_{q}$ linear code $\mathcal{C}_{h}$, whose $σ$ dual code has minimum distance $d'$, such that there exist an $[[n,k-h,d;n-k-h]]_{q}$ EAQECC and an $[[n,n-k-h,d';k-h]]_{q}$ EAQECC for every integer $h$ with $0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))$. Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to $σ$ hulls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $σ$ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes Cao, Meng Information Theory Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the $σ$ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be $σ$ dual-containing and $σ$ self-orthogonal. We prove that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))$. We prove that for any integer $h$ with $\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})$, there exists a linear code $\mathcal{C}_{2,h}$ monomially equivalent to $\mathcal{C}_{2}$ such that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=h$, where $\mathcal{C}_{i}$ is an $[n,k_{i}]_{q}$ linear code for $i=1,2$. We show that given an $[n,k,d]_{q}$ linear code $\mathcal{C}$, there exists a monomially equivalent $[n,k,d]_{q}$ linear code $\mathcal{C}_{h}$, whose $σ$ dual code has minimum distance $d'$, such that there exist an $[[n,k-h,d;n-k-h]]_{q}$ EAQECC and an $[[n,n-k-h,d';k-h]]_{q}$ EAQECC for every integer $h$ with $0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))$. Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to $σ$ hulls. |
| title | The $σ$ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2405.07740 |