Investigation of automorphism group for code associated with optimal curve of genus three

The main result of this paper is contained in two theorems. In the first theorem, it is proved that the mapping $\lambda: \mathcal{L}(mP_\infty) \rightarrow \mathcal{L}(mP_\infty)$ has the multiplicative property on the corresponding Riemann — Roch space associated with the divisor $mP_\infty$ 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 $\lbrace -19, -43, -67, -163 \rbrace$ 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 $\lambda$, 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 $m \geq 4$ 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 $\sum\limits_{i=1}^nP_i$ and $mP_\infty$, where $P_i$ are points of the degree one.