Irreducible polynomials in the theory of cyclic codes play the role of generators of polynomials or polynomials. The irreducible polynomial itself is a divisor of the polynomial xn (+) 1 without remainder. cannot be decomposed into simpler polynomials. The idea of building a cyclic code is based on the concept of an irreducible polynomial, which is divided only into itself and into a unit, i.e. F1 (x) = x3 + x2 + 1, F2 (x) = x + 1.ġ) F1 (x) + F2 (x) = x3 + x2 + x (in the binary system 1 + 1 = 0), 2) F1 (x) * F2 (x) = (x3 + x2 + 1) (x + 1) = x3 + x3 + x3 + x2 + x + 1 = x4 + x2 + x + 1,ĭerived from given by an arbitrary number of cyclic shifts also belongsĬyclic code. The multiplication of vectors corresponds to the multiplication by the multiplication rule of polynomials, the division of vectors is the division of polynomials, and the operation “-“ is transformed into the operation “+” modulo. In this case, the addition of vectors involves the addition of polynomials, which is carried out as a sum modulo x of coefficients of the same name. The representation of binary vectors in the form of a polynomial allows you to move from actions on vectors to action on polynomials. X = 2, ai = 0.1.ĮXAMPLE: to represent a numerical sequence in the form of a polynomial F (x) = 1x3 (+) 0x2 (+) 0x (+) 1, F (x) = x3 (+) 1. ai, where i = 0, (n-1) are the digits of this number system. X - the base of the number system in which the code is built. It is convenient to set the cyclic binary code vector in the form of a polynomial (and not a combination of 0.1).
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