Error detection One of the coding theory aims is the detection and correction of errors arising in the code message And now we will show how detect errors in the code message by using the Fibonacci coding. The main idea is using the property of the matrix determinant as the check criterion of the transmitted message
Let's demonstrate this idea for the example of interpretation of the message
We know from the preceding pages of our Museum that the determinant of the matrix (2) can be calculated according to the following formula:
Let's consider now the determinant of the code message
Using (1) we can write the expression (4) in the following form:
The formula (5) shows that the determinants of the initial matrix (Det
If the number
The identities (5), (6), (7) underlay the basis of the new method of the error detection based on the application of the Fibonacci Let's demonstrate this method for the above-considered example. The determinant of the initial message is determined according to (3). We will use the Det Really, we have:
It means that the code message Let's consider now the numerical example considered in the preceding page of our Museum. Let's suppose that the initial matrix has the following form:
The determinant of (9) is equal to
Then after the Fibonacci coding by means of the multiplication by the coding matrix
Its determinant is equal to
Let's compare the calculation results (10) and (12). We can see that Det Let's consider the case when the code matrix (11) is destroyed in process of its transmission through the "communication" channel. Lets suppose the "destruction" of the matrix (11) consists of the fact that one of the entries differs from the initial value. For example the first entry 3319 takes the value 2076. Let's consider now the "destroyed" matrix
Let's calculate the determinant of the matrix (13):
Comparing (14) and (10) we can see that this douse not correspond to the fundamental condition (8) and hence the matrix (13) is not correct. However, the error detection is only the first step in communication of messages. We should correct the error in the code message. And we will show at the next page of our Museum how this can be made. Follow us! |