A new construction of Algebraic Geometry code using Trace function
Nupur Patanker, Sanjay Kumar Singh · arXiv (Cornell University) · 2020
In this note, we give a construction of Algebraic-Geometry 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 $\mathbb{F}_{q}$. We compute a bound on the dimension of this 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 $F$), $1 \leq r < \infty$. This code is a generalization of the geometric Goppa code, with no restriction on the length of the code except the support condition on divisors defining the code. We obtained few quasi-cyclic codes over $\mathbb{F}_{p}$ as examples of these codes.