On the genus of the curve corresponding to the subcode of low weight of a rational Goppa code

One of the main ways to provide correctness of information transmission via communication channels is the use of error-correcting codes. Construction of certain classes of codes is based on the curves with sufficient number of rational points. In this paper we study abelian curves.
According to algorithm of construction, first of all, it is necessary to represent subcode of low weight as a trace code. Let $C_L (D,aP_\infty)$ be a rational Goppa code over $F_p$ with parameters $[n, k]$ and let $D_r$ denote the r-dimensional subcode of this code such that

$\left| {\chi (D_r )} \right| = d_r (C_L (D,aP_\infty ))$.

We need to represent subcode of low weight as follows

$Tr_{Con(D)} (U) = \left\{ {Tr_{Con(D)} (R)\left| {R \in U} \right.} \right\} = D_r $,

where $U$ is $r$-dimensional $F_p$-vector space and $Tr$ is trace map

$Tr:F_{p^m } \to F_p $.

Let $E_U$ be the function field of curve $C_{D_r}$, corresponding to the subcode of low weight $D_r$. So, the curve over field $F_{p^m} $ corresponds to the subcode of low weight. The genus of this curve is

$g(C_{D_r } ) = \sum\limits_{i = 1}^t {g(E_i )}$, $ t=\frac{p^r-1}{p-1}$,