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 ⎜⎜⎜⎝ 𝑔( 1)-1 𝑔( 2)-1 . . . 𝑔( 𝑛)-1 1𝑔( 1)-1 2𝑔( 2)-1 . . . 𝑛𝑔( 𝑛)-1 ... ... ... 𝑡-1 1 𝑔( 1)-1 𝑡-1 2 𝑔( 2)-1 . . . 𝑡-1 𝑔( 𝑛)-1 ⎟⎟⎟⎠ . Elements 1, ..., 𝑛 are elements of the finite field 𝐹𝑞𝑚. We define a set = { 1, ..., 𝑛} ⊆ 𝐹𝑞𝑚, |𝐿| = 𝑛. Polynomial 𝑔(𝑥) ∈ 𝐹𝑞𝑚[𝑥] is a polynomial of degree such that 1 ≤ ≤ - 1 and 𝑔( 𝑖) ̸= 0 for all elements 𝑖 ∈ 𝐿. Let 𝐺0 be the zero divisor of polynomial 𝑔(𝑥) in the divisor group of the rational function field 𝐹𝑞𝑚(𝑥). Let denote the zero of (𝑥 - 𝑖) for all 𝑖 ∈ 𝐿. We define the divisor as the sum of places of degree one = 𝑃1 + 𝑃2 + ... + 𝑃𝑛. Thus foregoing matrix generate a rational Goppa code 𝐶𝐿(𝐷𝐿,𝐺0 - 𝑃∞). 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 𝐶𝐿(𝐷𝐿,𝐺0 - 𝑃∞) is defined by formula 𝑑𝑟(𝐶𝐿(𝐷𝐿,𝐺0 - 𝑃∞)) = - + 𝑟, где 1 ≤ ≤ 𝑘. Let denote the r-dimensional subcode of code 𝐶𝐿(𝐷𝐿,𝐺0 - 𝑃∞) of low weight. Elements 𝑓1, . . . , of vector space 𝐿(𝐷𝐿,𝐺0 - 𝑃∞) correspond to elements 𝑒𝑣𝐷(𝑓1), . . . , 𝑒𝑣𝐷(𝑓𝑟) of basis of code 𝐷𝑟. Condition |𝜒(𝐷𝑟)| = 𝑑𝑟(𝐶𝐿(𝐷𝐿,𝐺0 - 𝑃∞)) determines the structure of the principal divisors (𝑓𝑖) (𝑓𝑖) = + - (𝐺0 - 𝑃∞), 1 ≤ ≤ 𝑟. At the same time, the divisors and satisfy the requirements 0 ≤ ≤ 𝐷𝐿, deg𝐷 = - and ≥ 0, deg𝐵𝑖 = - 1 для 1 ≤ ≤ 𝑟.