Question

A Hamming code is a technique where errors can not only be detected but can also be corrected. The simplest example of this kind of code is the (7,4)-Hamming code. In this scheme, a codeword is 7 bits long. We number the positions as follows:

                1   2   3   4   5   6   7

The message that is sent is only four bits long, with these four bits occupying positions 3, 5, 6, and 7. Bits 1, 2, and 4 are "parity check" bits as follows:

                Bit 1 checks positions 1, 3, 5, and 7. That is, the bit for position 1 is chosen in such a way that the total number of 1's in positions 1, 3, 5, and 7 is even.

                Bit 2 checks positions 2, 3, 6, and 7.

                Bit 4 checks positions 4, 5, 6, and 7.

For example, if I wanted to send the message   0 1 0 0,    I would encode it as follows:

                                                1   0   0   1   1   0   0

A. How would you encode the following messages:                    0 1 0 1

                                                                                                                                                                0 0 1 0

B. What messages did the sender intend to send you if you receive:

                                0 1 1 1 1 0 0

                                1 1 1 1 1 1 0

Answer 1

Data (B1, B2, B3, B4) Transmitted Code (x1, x2, B1, x3, B2, B3, B4) B1 B2 B3 B4 X1 X2 B1 X3 B2 B3 B4 010 1 Part 1: B1+B2+B4 0+1+1-2- (even) 0Ex1 Part 2: B1 +B3tB4-0+0+1#1-(odd) 15x2 Part 3: B2+B3+B4-1+0+1-2(even) 0 X3 00 1 0 Part 1: B1+B2+B4 0+0+0 0 (even) 0Ex1 Part 2: B1 +B3+B4-0+1 +0-1-(odd) 1-x2 Part 3: B2+B3+B4 0+1+0-1 (odd) 1Ex3

Transmitted Code (x1, x2, B1, x3, B2, B3, B4) Data (B1, B2, B3, B4) 01 11 1 00 11 00 11 1 0