Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices.

Abstract

In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking $ x = 1 $. We call this coding theory the k-order Gaussian Fibonacci coding theory. This coding method is based on the $ {Q_k}, {R_k} $ and $ E_n^{(k)} $ matrices. In this respect, it differs from the classical encryption method. Unlike classical algebraic coding methods, this method theoretically allows for the correction of matrix elements that can be infinite integers. Error detection criterion is examined for the case of $ k = 2 $ and this method is generalized to $ k $ and error correction method is given. In the simplest case, for $ k = 2 $, the correct capability of the method is essentially equal to 93.33%, exceeding all well-known correction codes. It appears that for a sufficiently large value of $ k $, the probability of decoding error is almost zero.