The conorm code of an AG-code

TL;DR

This paper introduces the conorm code construction for algebraic geometry codes over function field extensions, analyzing their parameters, duality properties, and connections to classical codes like Reed-Solomon and Hermitian codes.

Contribution

It defines the conorm code for AG-codes, studies its parameters and duality, and shows how classical codes can be represented as conorm codes.

Findings

Conorm codes preserve duality under certain conditions.

Repetition, Hermitian, and Reed-Solomon codes can be expressed as conorm codes.

Parameter analysis relates conorm codes to base AG-codes and extension properties.

Abstract

Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\text{Con}_{F'/F}(\mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\text{Con}_{F'/F}(\mathcal{C})$ in terms of the parameters of $\mathcal{C}$, the ramification behavior of the places used to define $\mathcal{C}$ and the genus of $F$. In the case of unramified extensions of function fields we prove that $\text{Con}_{F'/F}(\mathcal{C})^\perp = \text{Con}_{F'/F}({\mathcal{C}}^\perp)$ when the degree of the extension is coprime to the characteristic of $\mathbb{F}_q$. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.