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arXiv · 1910.10753

The conorm code of an AG-code

María Chara·Ricardo A. Podestá·Ricardo Toledano
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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.

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María Chara, Ricardo A. Podestá, Ricardo Toledano. 2019-10-23. The conorm code of an AG-code. https://arxiv.org/abs/1910.10753

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