Extended Hamming Codes

Hello!

People, I need your help!!!!!!

For example, I have a parity-check matrix for (7,4). Let it be:

% Parity-Check MAtrix
H = [
1 0 1 1 1 0 0;
1 1 1 0 0 1 0;
0 1 1 1 0 0 1
];

% Generator MAtrix
G = [
1 0 0 0 1 1 0;
0 1 0 0 0 1 1;
0 0 1 0 1 1 1;
0 0 0 1 1 0 1
];

% Uncoded Message
msg = [1 1 0 0];

% Coded Message
coded_msg = rem(msg * G, 2);

% Coded Message With no Error
coded_msg_err = [1 1 0 0 1 0 1];

syndrome = rem(coded_msg_err * H', 2)

syndrome = 0 0 0 (no errors).

Now I want to add a parity bit. I want (8,4) code. Ext. HAmming.

How should I change my H matrix?

Like this?
H = [
1 0 1 1 1 0 0 0;
1 1 1 0 0 1 0 0;
0 1 1 1 0 0 1 0;
1 1 1 1 1 1 1 1
];

Is it correct?

Please help me.

If the minimum distance of the modified H is 4, I think it is correct.

what are the advantages of (8,4) over (7,4) if it still can correct no more than 1 error (as (7,4))?

t=[dmin-1/2]

[(4-1)/2] = [(3-1)/2].

if I use (7,4) code I can identify the position of 1 error using the value of syndrome.
is it the same with (8,4)?

-thx in advance

The error-detecting capability is increased ( dmin-1 ).
Maybe some of 2-bits error patterns can be corrected, but i'm not sure.
( try to create the standard array of (8,4) code )

Think your H matrix is OK.

Well, (8,4) doesnt correct more bits, but it can detect 2 bit errors. (7.4) cannot do that.
-b