Coding theory on generalized Fibonacci n-step polynomials

Abstract

In this paper, we introduce generalized Fibonacci n-step polynomials. Based on generalized Fibonacci n-step polynomials, we define a new class of square matrix M_{h, n}(x) of order n. Thereby, we state a new coding theory called generalized Fibonacci n-step polynomial coding theory. Also we obtain the relations among the code matrix elements. Next, we discuss on the correct ability of this method. It is shown that, for n = 2, the correct ability of this method is 93.33% whereas for n = 3, the correct ability of this method is 99.80%. The interesting part of this coding/decoding method is that the correct ability does not depend on the polynomial coefficients excepting the value of the nth coefficient h_n(x) = 1 and increases as n increases.

Keywords:

• Fibonacci numbers
• Fibonacci n-step numbers
• Fibonacci polynomials
• Fibonacci n-step polynomials
• generalized Fibonacci n-step polynomials
• generalized Fibonacci n-step polynomial coding
• error correction