This question is about the process of using matrices to implement the Hamming (15, 11) code (in the form given in the lecture notes). You should explicitly give all the matrices used in the process. You may use Excel or other software to carry out the matrix operations, if you wish, but you should give the results in full and include your Excel formulas or code.

Create an 11-digit message by writing the last six digits of your enrolment number followed by the digits 01010. So for example, with enrolment number 001234567 we would create 23456701010. Now create an eleven-bit message by converting each of these digits to the equivalent (mod 2), so even numbers are written as 0 and odd numbers as 1 this example would give 01010101010.

(a) What is the distance of this code? What are its message length, block length and rate? How many errors can it detect, and how many can it correct?

[1 MARK]

(b) Write down the generating matrix for this code and use it to encode your message. [2 MARKS]

(c) Write down the parity check matrix and the recovery matrix. [2 MARKS]

(d) To show how an error is corrected, introduce an error into your codeword by flipping the bit in position (n + 6), where n is the final digit of your enrolment number (that is, if the bit was 0, change it to 1, and if it was 1, change it to 0). Assume that you have received this string and show how the parity check matrix can be used to show that an error has occurred and to determine which bit is incorrect. Correct the error and recover the intended message.

[5 MARKS]

(e) Show what happens in the decoding if the first and second bits of your encoded message are both changed, and comment on your answer. [2 MARKS]