Исследование группы автоморфизмов кода, ассоциированного с оптимальной кривой рода три

Прикладная дискретная математика. 2022. № 56. С. 5–16.

Malygina E.

The main result of this paper is contained in two theorems. In the first theorem, it is proved that the mapping λ : L (mP∞) → L (mP∞) has the multiplicative property on the corresponding Riemann — Roch space associated with the divisor mP∞ which defines some algebraic-geometric code if the number of points of degree one in the function field of genus three optimal curve over finite field with a discriminant {−19, −43, −67, −163} has the lower bound 12m/(m−3). Using an explicit calculation with the valuations of the pole divisors of the images of the basis functions x, y, z in the function field of the curve via the mapping λ, we have proved that the automorphism group of the function field of our curve is a subgroup in the automorphism group of the corresponding algebraic-geometric code. In the second theorem, it is proved that if m > 4 and n > 12m/(m − 3), then the automorphism group of the function field of our curve is isomorphic to the automorphism group of the algebraic-geometric code associated with divisors Pn i=1 Pi and mP∞, where Pi are points of the degree one.

Keywords: алгеброгеометрический код, Function field, Algebraic-geometric code, optimal curve, automorphism group of AG-code, оптимальная кривая, функциональное поле, группа автоморфизмов код