The conorm code of an AG-code

Abstract

Given a suitable extension $F^{\prime }/F$ of algebraic function fields over a finite field ${\mathbb{F}}_q$, we introduce the conorm code ${\text{Con}}_{F^{\prime }/F}(\mathcal{C})$ defined over $F^{\prime }$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of ${\text{Con}}_{F^{\prime }/F}(\mathcal{C})$ 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 ${\text{Con}}_{F^{\prime }/F}(\mathcal{C}{)}^{\perp }={\text{Con}}_{F^{\prime }/F}({\mathcal{C}}^{\perp })$ 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.

arXiv commentTo appear in Advances in Mathematics of Communications