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

E. S. Malygina

Prikladnaâ diskretnaâ matematika, no. 2 (2022), pp. 5-16

Résumé

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.

Détail

Keywords: optimal curve, algebraic-geometric code, function field
Mots-clés : automorphism group of AG-code.