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# The conorm code of an AG-code

• * Corresponding author: María Chara
Partially supported by CONICET, UNL CAI+D 2016, SECyT-UNC, CSIC
• Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\operatorname{Con}_{F'/F}( \mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\operatorname{Con}_{F'/F}( \mathcal{C})$ 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 $\operatorname{Con}_{F'/F}( \mathcal{C})^\perp = \operatorname{Con}_{F'/F}( \mathcal{C}^\perp)$ 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.

Mathematics Subject Classification: Primary: 94B27; Secondary: 94B15, 14H05.

 Citation:

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