On the constuction of subcodes of low weight of a rational Goppa code

We study а class of rational Goppa codes which is closely related to classical Goppa codes. Classical Goppa codes were described by V.D. Goppa as a new class of error-correcting codes in 1970. At first, it was proposed a class of binary linear codes. The main idea was to set correspondence between the original set of binary vectors and the set of rational functions. One year later V.D. Goppa summarized the results and described the method of construction of $q$-ary error-correcting codes.
We consider a code defined by the generator matrix
$$ \begin{pmatrix} g(\alpha _1 )^{ - 1}& g(\alpha _2 )^{ - 1} &\ldots &g(\alpha _n )^{ - 1}\\ \alpha _1 g(\alpha _1 )^{ - 1}& \alpha _2 g(\alpha _2 )^{ - 1} &\ldots & \alpha _n g(\alpha _n )^{ - 1}\\ \vdots& \vdots & {} & \vdots\\ \alpha _1 ^{t - 1} g(\alpha _1 )^{ - 1}& \alpha _2 ^{t - 1} g(\alpha _2 )^{ - 1} &\ldots & \alpha _n ^{t - 1} g(\alpha _n )^{ - 1} \end{pmatrix} $$

Elements $\alpha _1 ,...,\alpha _n$ are elements of the finite field $ F_{q^m }$. We define a set $L = \{ \alpha _1 ,...,\alpha _n \} \subseteq F_{q^m } ,\quad \left| L \right| = n.$ Polynomial $g(x) \in F_{q^m }[x]$ is a polynomial of degree $t$ such that $1\leq t \leq n-1$ and $g(\alpha _i ) \ne 0$ for all elements $ \alpha _i \in L$.
Let $G_0$ be the zero divisor of polynomial $g(x)$ in the divisor group of the rational function field $ F_{q^m }(x)$. Let denote $P_i$ the zero of $(x-\alpha _i)$ for all $ \alpha _i \in L$. We define the divisor $D_L $ as the sum of places of degree one

$D_L = P_1 + P_2 + ... + P_n$.

Thus foregoing matrix generate a rational Goppa code $C_L (D_L,G_0-P_\infty)$.
In this paper, we study structure of subcodes of low weight of such rational Goppa codes. We analyze, in the term of divisors, construction of subcodes of low weight. Our analysis is based on the knowledge of weight hierarchy of codes. The weight hierarchy of code $C_L (D_L,G_0-P_\infty)$ is defined by formula

$d_r(C_L (D_L,G_0-P_\infty))=n-k+r$, где $1\leq r \leq k$.

Let $D_r$ denote the r-dimensional subcode of code $C_L (D_L,G_0-P_\infty)$ of low weight. Elements $f_1,…, f_r$ of vector space $L(D_L,G_0-P_\infty)$ correspond to elements $ev_D(f_1),…, ev_D(f_r)$ of basis of code $D_r$. Condition

$|\chi (D_r )| = d_r (C_L (D_L,G_0-P_\infty))$

determines the structure of the principal divisors $(f_i)$

$(f_i ) = D + B_i - (G_0 - P_\infty )$, $ 1\leq i \leq r$.

At the same time, the divisors $D$ and $B_i$ satisfy the requirements

$ 0 \le D \le D_L$, $\deg D = t - r$

and

$B_i \ge 0$, $\deg B_i = r - 1$ для $ 1\leq i \leq r$.