Anyone here know about hamming codes, as they are used to detect and correct transmission errors?

I've got a maths exam later and this is the only topic I don't feel I know well enough.

I've got the stuff about hamming distance - d+1 and 2d+1 for transmission error detection and correction.

From there it all kind of falls apart.

If the hamming code is (7,4) then you get k1, k2, k3, and k4 for the info bits, and c1,c2,c3 for the correction bits right, because 7-4=3 so 3 correction bits? If it were (9,3) then you'd get k1, k2, k3 and c1,c2,c3,c4,c5,c6 because 9-3 = 6 correction bits ?

But from there I'm not sure how to create the matrix. I've been told that the matrix for (7,4) is:

k1 k2 k3 k4 - c1 c2 c3
0 0 0 0 - 0 0 0
0 0 0 1 - 1 1 1
0 0 1 0 - 0 1 1
0 0 1 1 - 1 0 0
0 1 0 0 - 1 0 1
0 1 0 1 - 0 1 0
0 1 1 0 - 1 1 0
0 1 1 1 - 0 0 1
1 0 0 0 - 1 1 0
1 0 0 1 - 0 0 1
1 0 1 0 - 1 0 1
1 0 1 1 - 0 1 0
1 1 0 0 - 0 1 1
1 1 0 1 - 1 0 0
1 1 1 0 - 0 0 0
1 1 1 1 - 1 1 1

WTF? I can't see any patterns or reason in the c1 c2 c3 numbers. I can see the K1 k2 k3 k4 part - just counting out all the possible values from 0-15 because it's four information bits.

But how do we calculate the c1 c2 c3 values from the k1 k2 k3 k4 values?