Show that this code can correct double errors. What is the result of encoding the messa...

Show that this code can correct double errors.

What is the result of encoding the messages in using the (7, 4) Hamming code of Example 3.71?

Reference Example 3.71

Suppose we want to design an error-correcting code that uses three parity check equations. Since these equations give rise to the rows of P, we have n – k = 3 and k = n – 3. The message vectors come from One such candidate is

which we recognize as column 3 of P. Therefore, the error is in the third component of c ′, and by changing it we recover the correct code vector c.We also know that the first four components of a code vector are the original message vector, so in this case we decode c to get the original x = [0 1 0 1]^T

The code in Example 3.71 is called the (7, 4) Hamming code. Any binary code constructed in this fashion is called an (n, k) Hamming code. Observe that, by construction,

an (n, k) Hamming code has n = 2^n–k – 1.