Error correction Let's show now a possibility of the code message E restoration by using the properties of the Q-matrix determinant. Let's continue the consideration of the above example when the initial massage is represented in the matrix form:
The determinant of the matrix (1) is equal:
Let's suppose that the matrix Q^{5} is chosen for the Fibonacci coding. Then, the code matrix E takes the following form:
The essence of the code matrix restoration consists of the following. After constructing the code matrix E it is calculated the determinant of the initial matrix of M according to (2). The determinant is sent to the communication channel after the code message Let's assume that the communication channel has the special means for the error detection in each of elements of the code message E. Let's assume that the first element of E is received with the error. Then, we can represent the code message in the matrix form:
where x is the "destroyed" element of the code message E, but the rest matrix entries must be correct and equal to the following:
Then, according to the properties of the Fibonacci coding method for the case of the coding matrix Q^{5} the determinant of the matrix Ì' has to be equal to the determinant (2) with the opposite sign. Then, we can write the following equation for calculation of x:
After execution of the elementary transformations in the equation (6), we can get the solution of the equation in the form:
Comparing the calculated value (7) with the entry of the code matrix E given with (3) we can do the following important conclusion: x = . Thus, we have restored the code message E using the property of the Q-matrix determinant. But in the real situation usually we do not know what element of the code message is "destroyed". Let's consider the numerical example from the preceding pages of our Museum when the coding matrix Q^{5}, the initial matrix M, and the code matrix E have the following forms respectively:
and there exists the following correlation for the determinants:
Let's suppose that after comparison of Det E and Det M we have found that it douse not correspond to (11) that is the indication of the error in the code matrix (10). But we do not have the information about the "destroyed" elements of the matrix (10). In this case we can suppose different hypotheses' about the possible "destroyed" element and then we can test these hypotheses'. However we have one more condition for the elements of the code matrix E: all its elements are integers. That is why we can choose as valid only those results, which bring us into integer numbers! At least if our tests do not give integer numbers in any test this means that the error arose in two, three or four elements of our code matrix and this error is not correctable. And now we total our consideration of the new coding method based on the Fibonacci Q-matrix. We have considered unusual coding method! In our method the elements of the code matrix E are objects of the error detection and correction! We correct not separate bits or their combinations in the binary codes as in the classical code theory. We used the unusual mathematical apparatus for our coding theory, namely the matrix theory and the Fibonacci numbers theory. And we have showed that we can correct the matrix elements, which can be the numbers of the huge value (for example the binary numbers of the length by 10^{6} or 10^{12} bits that is impossible in the classical coding theory)! That is way the method considered above is of the principal importance for the modern theory of correcting codes. And if you will be interested in our theory of coding, we are ready to cooperate to you in the field of software and hardware implementation of the Fibonacci coding method. Address to us! |