The Dimensions of the Hulls of Conorm Codes from Algebraic Geometry Codes

Abstract

Chara et al. introduced conorm codes defined over algebraic geometry codes, but the hulls of conorm codes were not determined yet. In this paper, we study the dimension of the hull of conorm codes using the method introduced by Camps et al. For an algebraic geometry code $\mathcal{C}:=C_\mathscr{L}(D, G)$, we consider the divisor $\gcd(G, H)$, where $H$ is the divisor satisfying \[C_\mathscr{L}(D, G)^\perp=C_\mathscr{L}(D, H).\] Given an extension $F'/\mathbb{F}_{q^t}$ of an algebraic function field $F/\mathbb{F}_q$, we assume that the divisor $\gcd(G, H)$ is non-special. If the degree of $\gcd(G, H)$ is greater than $2g-2+{t\over [F':F]}\deg\text{Diff}(F'/F)$, then we have determined the exact dimension of the hull of the conorm of $\mathcal{C}$. If not, we have determined the lower bound of the dimension of the hull of the conorm of $\mathcal{C}$. We provide some examples for the dimension of the hull of certain conorm codes of AG codes defined over a rational function field.