A Fibonacci-polynomial based coding method with error detection and correction

Abstract

a b s t r a c tA Fibonacci coding method using Fibonacci polynomials is introduced. For integers m ≥ 2, x ≥ 1 and n ≥ 1, an m × m matrix Q n m (x), the nth power of Q m (x), is considered as the encoding matrix, where Q m is an m × m matrix whose elements are Fibonacci polynomials.The decoding matrix Q −n m (x) is also introduced. A simple error-detecting criterion and a simple error-correcting method for this class of codes are given. It is shown that the probability of decoding error is almost zero for m large enough. Illustrative examples are provided.

High rate fibonacci polynomial codes

“…The basic idea behind the encoding and decoding algorithms given in this section is the same as in [10]. The main difference is the change of encoding/decoding matrices in order to obtain higher code rates.…”

“…This sequence has been extended in many ways, two of which have been used in coding theory. These are known as the p-Fibonacci sequence [9], and the Fibonacci polynomials [10]. The p-Fibonacci sequence is defined by the recurrence relation F p (n) = F p (n − 1) + F p (n − p − 1), n > p + 1, p ≥ 0, F p (1) = F p (2) = F p (3) = .…”

“…In [10], square matrices Q m (x) of order m ≥ 2, consisting of Fibonacci polynomials as their elements, were introduced.…”

“…The matrix Q n m (x) was employed in [10] to provide a channel coding technique. Compared to the Q p matrices, this method has the advantage of an infinite number of encoding/decoding matrices of a constant order and power.…”

The Extended Golden Section and Time Series Analysis

“…This marked the first visual construction connecting the Golden and Silver mean proportions in a single diagram [18]. Today the Golden ratio plays an increasing role in engineering and modern physical research [19,20,21,22,23,24,25,26,27,28,29]. Some of its applications in signal processing include:…”