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 𝐶𝐿(𝐷, 𝑎𝑃∞) be a rational Goppa code over with parameters [n, k] and let denote the 𝑟-dimensional subcode of this code such that |𝜒(𝐷𝑟)| = 𝑑𝑟(𝐶𝐿(𝐷, 𝑎𝑃∞)). We need to represent subcode of low weight as follows TrCon(𝐷)(𝑈) = {︀TrCon(𝐷)(𝑅) |𝑅 ∈ }︀= 𝐷𝑟, where is 𝑟-dimensional 𝐹𝑝-vector space and Tr is trace map Tr : → 𝐹𝑝. Vector space can be constructed in the following way. Let {𝑐1,..., 𝑐𝑟} be a basis of subcode of low weight of a rational Goppa code. Elements 𝑅1,...,𝑅𝑟 correspond to elements of basis and can be constructed as 𝑅𝑓𝑖(𝑥) = ( 𝑚-1 Σ︁𝑠=0 (𝑏𝑥)𝑝𝑠 )𝑎-1𝑏𝑥 - Σ︁𝑗=1 𝑖𝑗( 𝑚-1 Σ︁𝑠=0 (𝑏𝑥)𝑝𝑠 )𝑎-2𝑏𝑥 + + Σ︁𝑗̸=𝑘 𝑖𝑘( 𝑚-1 Σ︁𝑠=0 (𝑏𝑥)𝑝𝑠 )𝑎-3𝑏𝑥 - · · · + (-1)𝑎-2 Σ︁ 𝑗1