Исследование группы автоморфизмов кода, ассоциированного с оптимальной кривой рода три

Abstract

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 n associated with divisors ∑ Pi и mP∞, where Pi are points of the degree one.