The Dimensions of the Hulls of Conorm Codes from Algebraic Geometry Codes
Junmin An, Jon-Lark Kim
TL;DR
This work determines the hull dimensions of conorm codes derived from algebraic geometry codes by applying the Camps et al. framework. For $\mathcal{C}=C_{\mathscr{L}}(D,G)$ with dual $C_{\mathscr{L}}(D,H)$ and non-special $\gcd(G,H)$, a degree threshold involving $\deg \gcd(G,H)$ and the different divisor $\text{Diff}(F'/F)$ yields exact hull dimensions under unramified extensions, while ramified extensions give precise lower bounds and equalities under a corresponding threshold. The authors also analyze conorms of rational AG codes, providing explicit hull formulas for two-point rational codes and specialized results for elliptic, hyperelliptic, and Hermitian function fields. Additionally, the paper establishes that, in certain unramified scenarios, LCD codes remain LCD under conorm, and presents several concrete examples over rational function fields. These results advance the understanding of hulls in conorm codes, informing code design and potential cryptographic applications.
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 $2g-2+{t\over [F':F]}°\text{Diff}(F'/F)$, 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.