Final answer:
The message 1111001 was checked for errors using the Hamming code by calculating the parity bits and fixing any errors detected. The erroneous bit was identified, and the corrected message was determined to be 1110001. The actual data extracted from the corrected message is 1001.
Step-by-step explanation:
The student has received a message 1111001 encoded with the [7,4] binary Hamming code. To check for errors and decode the message, one must calculate the parity bits 'a', 'b', and 'c'. Here is how you can do it:
- a = x4 + x5 + x6 + x7 (mod 2)
- b = x2 + x3 + x6 + x7 (mod 2)
- c = x1 + x3 + x5 + x7 (mod 2)
Let's assume that the received message 1111001 corresponds to x7 x6 x5 x4 x3 x2 x1. We substitute the values into the parity equations:
- a = 0 + 0 + 1 + 1 (mod 2) = 0 (No error in the group of x4, x5, x6, x7)
- b = 0 + 1 + 1 + 1 (mod 2) = 1 (Error detected in the group of x2, x3, x6, x7)
- c = 1 + 1 + 0 + 1 (mod 2) = 1 (Error detected in the group of x1, x3, x5, x7)
The binary sequence for the error location is '011', which is equal to '3' in decimal. So, the error is in the third bit from the right (x5). Correcting the error, we flip the third bit, changing the message to 1110001. The original four-bit data message can then be extracted by removing the parity bits, which gives us 1001.