Higher weight spectra of ternary codes associated to the quadratic Veronese $3$-fold
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916252323151872 |
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| author | Kaipa, Krishna Pradhan, Puspendu |
| author_facet | Kaipa, Krishna Pradhan, Puspendu |
| contents | The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12011 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher weight spectra of ternary codes associated to the quadratic Veronese $3$-fold Kaipa, Krishna Pradhan, Puspendu Combinatorics Information Theory 94B27, 51E20, 05B25 The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work. |
| title | Higher weight spectra of ternary codes associated to the quadratic Veronese $3$-fold |
| topic | Combinatorics Information Theory 94B27, 51E20, 05B25 |
| url | https://arxiv.org/abs/2405.12011 |