Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

• Authors: Frédéric A. Edoukou
• Venue: Designs, Codes and Cryptography
• Year: 2009

We study the functional codes of second order on a non-degenerate Hermitian variety $${\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)}$$ as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of $${\mathcal{X}}$$ . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety $${\mathcal{X} \subset{\mathbb{P}}^4(\mathbb{F}_q)}$$ . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety $${\mathcal{X} \subset {\mathbb{P}}^N(\mathbb{F}_q)}$$ .