On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908729159450624 |
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| author | Sagar, Vidya Patel, Shikha Singh, Ashutosh Garani, Shayan Srinivasa |
| author_facet | Sagar, Vidya Patel, Shikha Singh, Ashutosh Garani, Shayan Srinivasa |
| contents | We consider two-dimensional $(λ_1, λ_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We also describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes from the generator tensor, implementable in a parallel fashion. Through examples, we illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate, generalizing prior work over 2-D binary cyclic coded arrays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets Sagar, Vidya Patel, Shikha Singh, Ashutosh Garani, Shayan Srinivasa Information Theory We consider two-dimensional $(λ_1, λ_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We also describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes from the generator tensor, implementable in a parallel fashion. Through examples, we illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate, generalizing prior work over 2-D binary cyclic coded arrays. |
| title | On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets |
| topic | Information Theory |
| url | https://arxiv.org/abs/2412.09915 |