Session S09 - Number Theory in the Americas

Wednesday, July 14, 14:00 ~ 14:30 UTC-3

The conorm code of an AG-code

Maria Chara

U. N. Litoral, Santa Fe, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For a given trascendental element $x$ over $\mathbb{F}_q$, the field of fractions of the ring $\mathbb{F}_q[x]$ is denoted as $\mathbb{F}_q(x)$ and it is called a rational function field over $\mathbb{F}_q$. An (algebraic) function field $F$ of one variable over $\mathbb{F}_q$ is a field extension $F/\mathbb{F}_q(x)$ of finite degree. The \textit{Riemann-Roch space} associated to a divisor $G$ of $F$ is the vector space over $\mathbb{F}_q$ defined as $$\mathcal{L}(G)=\{x\in F\,:\, (x)\geq G\}\cup \{0\},$$ where $(x)$ denotes the principal divisor of $x$. It turns out that $\mathcal{L}(G)$ is a finite dimensional vector space over $\mathbb{F}_q$ for any divisor $G$ of $F$. Given disjoint divisors $D=P_1+\cdots+P_n$ and $G$ of $F/\mathbb{F}_q$, where $P_1,\ldots,P_n$ are different rational places, the \textit{algebraic geometry code} (AG-code for short) associated to $D$ and $G$ is defined as \begin{equation} \label{CDG} C_\mathcal{L}^F (D,G) = \{(x(P_1),\ldots, x(P_n))\,:\,x\in \mathcal{L}(G)\}\subseteq (\mathbb{F}_q)^n, \end{equation} where $x(P_i)$ denotes the residue class of $x$ modulo $P_i$ for $i=1,\ldots,n$. In this talk the concept of the \textit{conorm code} associated to an AG-code will be introduced. We will show some interesting properties of this new code since some well known families of codes such as repetition codes, Hermitian codes and Reed-Solomon codes can be obtained as conorm codes from other more basic codes. We will see that in some particular cases over geometric Galois extensions of function fields, the conorm code and the original code are different representations of the same algebraic geometry code.

Joint work with R. Podesta (U. N. Cordova, Argentina) and R. Toledano (U.N. Litoral, Argentina).

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