k-order Gaussian Fibonacci polynomials and applications to the coding/decoding theory

Abstract

In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give the generating function, explicit formula and some identities for k-order Gaussian Fibonacci polynomials. We introduce the matrix repreŞent and we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a new coding theory called k-order Gaussian Fibonacci Polynomials coding theory and establish the code elements for values of k. This coding/decoding method bound to the Q(k) (x), R-k (x) and E-k,E-n (x) matrices. So, this method is different from the classical algebraic coding. Consequently, with this method, we move the coding theory onto a complex space which is a different field. Therefore, new working areas are created.