Quadratic Residue Codes over $\mathbb{Z}_{121}$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912982734209024 |
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| author | Chatterjee, Tapas Jain, Priya |
| author_facet | Chatterjee, Tapas Jain, Priya |
| contents | In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \) and \( p \equiv \pm 19 \pmod{44} \) over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and extended quadratic residue codes over $\mathbb{Z}_{121}$ are discussed, followed by their Gray images. Also, we show that the extended quadratic residue code over $\mathbb{Z}_{121}$ possesses a large permutation automorphism group generated by shifts, multipliers, and inversion, making permutation decoding feasible. As examples, we construct new codes with parameters $[55,5,33]$ and $[77,7,44].$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24689 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadratic Residue Codes over $\mathbb{Z}_{121}$ Chatterjee, Tapas Jain, Priya Information Theory Number Theory In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \) and \( p \equiv \pm 19 \pmod{44} \) over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and extended quadratic residue codes over $\mathbb{Z}_{121}$ are discussed, followed by their Gray images. Also, we show that the extended quadratic residue code over $\mathbb{Z}_{121}$ possesses a large permutation automorphism group generated by shifts, multipliers, and inversion, making permutation decoding feasible. As examples, we construct new codes with parameters $[55,5,33]$ and $[77,7,44].$ |
| title | Quadratic Residue Codes over $\mathbb{Z}_{121}$ |
| topic | Information Theory Number Theory |
| url | https://arxiv.org/abs/2603.24689 |