Fibonacci coding

We begin to develop new coding theory based on the -matrices since the simplest Fibonacci Q-matrix. Let's consider the following method of coding. Let's represent the initial message in form of the 2 ´ 2 matrix:

(1)

For example our message is the sequence of the decimal numerals:

Then we can represent our message (2) in the matrix form:

(3)

Suppose now that we have selected for coding the Fibonacci Q-matrix of the 5th power:

(4)

At the preceding pages of our Museum we have introduced the notion of the matrix "inverse" to (4). Because the number 5 is odd than the matrix "inverse" to (4) has the following form:

Then, the Fibonacci coding of the message M given in the matrix form (1) consists of the multiplication of the initial matrix (1) by the coding matrix (4), that is:

(5)

where

(6)

For our "unenlightened" reader we remember that the "matrix multiplication" is mathematical operation distinguished from the traditional "multiplication". We can see from the example (5) that the product of two square matrices M and Q⁵ is the matrix E of the same size, which elements are calculated according to (6).

Let's apply our calculations to our example (3). Then the procedure of the Fibonacci coding brings us into the following matrix E:

After that the code message is sent to the communication channel.

The decoding of the code message E is executed in the following manner. The code message E is represented in the matrix form and the code matrix E is multiplied by the inverse Q-matrix of the power of 5:

(7)

Let's calculate the entries of the matrix, which can be obtained after the decoding, taking into consideration (6):

Thus, we have:

Thus, we have showed a possibility to code and decode the initial numerical information by using the Fibonacci Q-matrix. However, our reader can give us the reasonable question: what is of the practical use of the Fibonacci coding? And we can answer this question at the next pages of our Museum. Follow us!