V. D. Goppa’s scientific contributions

Let X be an irreducible algebraic curve over the finite field F q , q = p m . Denote by S the finite set of the points of the curve X. If an integer n p > 0 is specified for every P ∈ S, then the module with support S is said to be defined. Thus the module m can be identified with the positive divisor ∑n p P. Let g be a rational function on X. If $vp\left( 1-g \right)\ge np,P\in S$ (4.1) then we use the notation $g\equiv 1\bmod \mathfrak{M}$ (4.2)

Let A be an arbitrary finite set (alphabet) consisting of $q = \left| A \right|$ letters. By A n we will denote the set of all sequences (words) of length n in the alphabet A. The Hamming distance ρ(x,y), with x, y belonging to A n , is the number of positions in which the words x and y are distinguished. It is easy to check that this function satisfies all distance axioms: $p\left( {x,y} \right) \geqslant 0$ (1) $p\left( {x,y} \right) = 0 = > x = y$ (2) $p\left( {x,z} \right) \leqslant p\left( {x,y} \right) + p\left( {y,z} \right)$ (3)

Let F(X. Y) be a polynomial of two variables over the field F q . A point (a, b) lying in the plane is called the root of the polynomial if F(a, b) = O. All these roots can be found by enumeration and define a plane affine curve. Usually, one considers all points with coordinates in the algebraic closure of the field F q , i.e., \( a,b \in {F_{{q^m}}},m = 1,2 \cdots . \) Points of the curve such that (a, b) ∈ F q are said to be rational over F q . A projective curve is defined as a set of points lying in the projective plane and nullifying the form F(X, Y, Z).

In the previous chapter we introduced normal rational codes and developed a method of decoding based on a simple geometric idea. The code vectors were described algebraically in terms of differential representation. Such a representation leads to a new point of view of the decoding.