Jacobi polynomials, invariant rings, and generalized $t$-designs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913687215800320 |
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| author | Chakraborty, Himadri Shekhar Hamid, Nur Miezaki, Tsuyoshi Oura, Manabu |
| author_facet | Chakraborty, Himadri Shekhar Hamid, Nur Miezaki, Tsuyoshi Oura, Manabu |
| contents | In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in genus $g$ with composition $T$ with $|T|\leq t$ can be obtained from its weight enumerator in genus~$g$ using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus $g$. Specifically, the generators of the invariant ring appearing for $g=1$ are obtained. Moreover, we define the split Jacobi polynomials in genus~$g$ and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a $t$-homogeneous code also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jacobi polynomials, invariant rings, and generalized $t$-designs Chakraborty, Himadri Shekhar Hamid, Nur Miezaki, Tsuyoshi Oura, Manabu Combinatorics Group Theory Number Theory In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in genus $g$ with composition $T$ with $|T|\leq t$ can be obtained from its weight enumerator in genus~$g$ using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus $g$. Specifically, the generators of the invariant ring appearing for $g=1$ are obtained. Moreover, we define the split Jacobi polynomials in genus~$g$ and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a $t$-homogeneous code also given. |
| title | Jacobi polynomials, invariant rings, and generalized $t$-designs |
| topic | Combinatorics Group Theory Number Theory |
| url | https://arxiv.org/abs/2408.02229 |