Investigation of automorphism group for code associated with optimal curve of genus three | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 56. DOI: 10.17223/20710410/56/1

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 ∑ P_i и mP_∞, where P_i are points of the degree one.