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

The theory of codes derived from algebraic curves was initiated by the works of V. D. Goppa. Since that time this theory has received an active development. Construction of certain classes of codes is based on the curves with sufficient number of rational points. In this paper we study curves arising from the subcode of low weight of a rational Goppa code.
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
$$\mathrm{Tr} _{\mathrm{Con}(D)} (U) = \left\{ {\mathrm{Tr} _{\mathrm{Con}(D)} (R)\left| {R \in U} \right.} \right\} = D_r,$$
where $U$ is $r$-dimensional $F_p$-vector space and $\mathrm{Tr}$ is trace map
$$\mathrm{Tr} :F_{p^m } \to F_p.$$

Vector space $U$ can be constructed in the following way. Let $\left\{c_{1},...,c_{r}\right\}$ be a basis of subcode of low weight of a rational Goppa code. Elements ${R_{1},...,R_{r}}$ correspond to elements of basis and can be constructed as

\begin{multline*} R_{f_i (x)} = (\sum\limits_{s = 0}^{m - 1} {\left( {bx} \right)^{p^s } } )^{a - 1} bx - \sum\limits_{j = 1}^a \alpha _{ij} (\sum\limits_{s = 0}^{m - 1} {(bx)^{p^s } } )^{a - 2} bx\ + \\ + \sum\limits_{j \ne k}^a {\alpha _{ij} \alpha _{ik} } (\sum\limits_{s = 0}^{m - 1} {(bx)^{p^s } } )^{a - 3} bx\ -\\ \cdots + ( - 1)^{a - 2} \sum\limits_{j_1 < \ldots < j_{a - 2} }^a {\alpha _{ij_1 } \cdot \ldots \cdot } \;\alpha _{ij_{a - 2} } \sum\limits_{s = 0}^{m - 1} {(bx)^{p^s } } bx +\\ +( - 1)^{a - 1} \sum\limits_{j_1 < \ldots < j_{a - 1} }^a {\alpha _{ij_1 } \cdot \ldots \cdot } \;\alpha _{ij_{a - 1} } bx + ( - 1)^a \alpha _i. \end{multline*}

Thus we obtain $R_1,...R_r \in F_{p^m}(x)$ such that $\mathrm{Tr} _{\mathrm{Con}(D)} (R_i ) = c_i,\;1 \le i \le r$, where $\left\{c_1,...,c_r\right\}$ is a basis of $D_r$.
We denote $ U = \left\langle {R_1, \ldots, R_r } \right\rangle $. Then $U$ is $r$-dimensional $F_p$-vector space and
$$\mathrm{Tr} _{\mathrm{Con}(D)} (U) = D_r.$$

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 equation of this curve is

$$\ y^{p^{r}}-y=\sum\limits_{j = 1}^r {\sum\limits_{i = 0}^{r-1} {\alpha^{j-1}}R^{p^{i}}_{j}}.$$