Linear Algebra 1:

We can interpret the parity check matrix as enforcing three constraints on our code words: 1. There are an even number of 1s in positions {4, 5, 6, 7} in the code word. 2. There are an even number of 1s in positions {2, 3, 6, 7} in the code word. 3. There are an even number of 1s in positions {1, 3, 5, 7} in the code word. Hence, the name parity check matrix makes sense (we are checking the parity of the sum of these three sets). Furthermore, if we look at the binary representations of 1 through 7, we see that the first set contains the numbers with a 1 in the third bit from the right (the 4s bit), the second set contains the numbers with a 1 in the second bit from the right (the 2s bit), and the third set contains the numbers with a 1 in the rightmost (1s) bit. Therefore, if our parity check vector Hc 6= 0, it tells us the (binary representation of the) index of the bit we need to flip to restore the proper parity.

Question: How would you go about trying to find patterns in the generator and parity check matrices for the (7, 4)-Hamming code. Use these to write down the matrices for the (15, 11)-Hamming code.