Анализ минимального расстояния АГ-кода, ассоциированного с максимальной кривой рода три

Прикладная дискретная математика. 2022. № 58. С. 5–14.

Malygina E., Кунинец А. А.

We consider a class of algebraic geometry codes associated with a maximal curve of genus three whose number of rational points satisfies the upper Hasse — Weil — Serre bound. It is proved that the number of rational points of such curve is odd and has a classification: the first type includes 4-tuples of conjugate points of multiplicity 1, the second type includes couples conjugate points of multiplicity 2, and the third type includes a single point of multiplicity 4. It is found out for which types of points the divisor of the functional field of the desired curve and consisting of these points is the principle. We consider special cases when deg(G) = 2, 4, and establish the form of a divisor D when AG-code CL (D, G) associated with the divisors D and G is MDS-code. It is shown that the AG-code CL (D, G) is not an MDS-code if the divisor D − G is principle and deg(G) > 5. Also, it is proved that CL (D, G) is an MDS-code if the divisor D consists only of the first type points of curve conjugated to each other for deg(D) > 8 and G = deg(D) + 2 2 P∞. Finally, it is shown that the dual equivalent code CL (D, H) ⊥ to the code CL (D, G), which is not MDS, will also not be MDS with conditions deg(D) − α < deg(H) < deg(D), 4 < deg(G) < α + 4, 5 < α < deg(D) − 5, and D consists only of conjugate points of the first type.

Keywords: алгеброгеометрический код, divisor, minimal distance, Function field, algebraic-geometry code, mds-code, maximal curves, минимальное расстояние кода, mds-код, максимальная кривая, функциональное поле, дивизор