Codes defined by forms of degree 2 on quadric and non-degenerate hermitian varieties in $\mathbb{P}^{4}(\mathbb{F}_q)$}
| dc.creator | Edoukou, Frederic A. B. | |
| dc.date | 2006-12-09 | |
| dc.date.accessioned | 2026-07-07T07:34:46Z | |
| dc.date.available | 2026-07-07T07:34:46Z | |
| dc.description | We study the functional codes of second order defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$ a quadric of rank($\mathcal{X}$)=3,4,5 or a non-degenerate hermitian variety. We give some bounds for %$# \mathcal{X}_{Z(\mathcal{Q})}(\mathbb{F}_{q})$, the number of points of quadratic sections of $\mathcal{X}$, which are the best possible and show that codes defined on non-degenerate quadrics are better than those defined on degenerate quadrics. We also show the geometric structure of the minimum weight codewords and estimate the second weight of these codes. For $\mathcal{X}$ a non-degenerate hermitian variety, we list the first five weights and the corresponding codewords. The paper ends with two conjectures. One on the minimum distance for the functional codes of order $h$ on $\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$ a non-singular hermitian variety. The second conjecture on the distribution of the codewords of the first five weights of the functional codes of second order on $\mathcal{X} \subset {\mathbb{P}}^N(\mathbb{F}_q)$ the non-singular hermitian variety. | |
| dc.description | submitted | |
| dc.identifier | https://arxiv.org/abs/math/0612229 | |
| dc.identifier | http://arxiv.org/abs/math/0612229 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/119886 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | 05B25, 11T71, 14J29 | |
| dc.title | Codes defined by forms of degree 2 on quadric and non-degenerate hermitian varieties in $\mathbb{P}^{4}(\mathbb{F}_q)$} | |
| dc.type | text |