The conorm code of an AG code

CHARA, MARÍA; PODESTÁ, RICARDO A.; TOLEDANO, RICARDO

ADVANCES IN MATHEMATICS OF COMMUNICATIONS

AMER INST MATHEMATICAL SCIENCES

Lugar: springfield; Año: 2023 vol. 17 p. 714 - 732

1930-5346

Given a suitable extension $F´/F$ of algebraic function fields over a finite field $F_q$, we introduce the conorm code $Con_{F´/F}(C)$ defined over $F´$ which is constructed from an algebraic geometry code $C$ defined over $F$. We study the parameters of $Con_{F´/F}(C)$ in terms of the parameters of $C$, the ramification behavior of the places used to define $C$ and the genus of $F$.In the case of unramified extensions of function fields we prove that $Con_{F´/F}(C)^perp = Con_{F´/F}(C^perp)$ when the degree of the extension is coprime to the characteristic of $F_q$. 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.