Generalization of trace codes to places of higher degree

Abstract

In this note, we give a construction of codes on algebraic function field $F/ \mathbb{F}_{q}$ using places of $F$ (not necessarily of degree one) and trace functions from various extensions of $\mathbb{F}_{q}$. This is a generalization of trace code of geometric Goppa codes to higher degree places. We compute a bound on the dimension of this code. Furthermore, we give a condition under which we get exact dimension of the code. We also determine a bound on the minimum distance of this code in terms of $B_{r}(F)$ ( the number of places of degree $r$ in $F$), $1 \leq r < \infty$. Few quasi-cyclic codes over $\mathbb{F}_{p}$ are also obtained as examples of these codes.