00:01
In this question, we need to construct a binary code c of length 8.
00:05
Okay, we have been given some equation, w is equal to a plus b plus c.
00:13
Then i have x equals to a plus b plus d, y is equal to a plus c plus d, and z is equal to b plus c plus d.
00:25
Okay, and let me write here that this four tuple, belongs to f2, 4 and a, b, c, d, w, x, y, z, which is the 8 digit code, it belongs to c.
00:45
In the first part, we need to show that c is a linear code.
00:52
So let us see how we are going to show this.
00:57
Okay, so if i consider here, so wx, z is what we have been given here.
01:06
Here i will put the value of wx y and z.
01:09
Okay, so i will get a, b, c, d and w will be what in this case, a plus b plus c, then x is given to be a plus b plus d, then a plus c plus d.
01:24
And then b plus c plus d.
01:28
This is what we have been given.
01:29
So, and this is belongings to c.
01:33
Correct.
01:35
This belongs to c.
01:37
Thing you can see in terms of addition of a, b, c and d.
01:41
So we can say that it is clearly a linear code.
01:49
Now let's move to the next part.
01:51
In the next part what we have to check is find a generator matrix and a parity check matrix for c.
02:00
So generator and the parity check matrix we have to find.
02:03
So what i will do? i will consider w1 as a1b1.
02:09
1 c1 then i will check a x1 i will take as a 1 b1 d1 similarly y 1 will be a 1 c1 d1 and then z 1 will be b1 c1 d1 correct and similarly i will consider for w2 as well a 2 b2 c2 then x2 i will take as a 2 b2 d2 then y 2 i will take as a 2 b2 d2 then y 2 i will take as a2, c2, d2, and z2, i will assume it to be b2, c2, d2, correct.
02:48
Now what else we can do here? i will consider, let, okay, x will be considered as a1, b, 1, c1, d1, and here what we are having, w1, x1, y, 1, z1.
03:11
And why will i assume as a2, b, 2, c2, d2, and then we have w2, x2, y, 2, and z2, correct? and we know that this both are the code.
03:30
Okay, then for some scalers, scale us let's say they are alpha and beta alpha x plus beta y that also going to belongs to this c1 plus c2 okay and alpha x will also belongs to c1 so this is belonging to i would say c1 and this belongs to c2 okay so alpha x belongs to this and beta y sorry i will take alpha on okay, alpha x and alpha y that is belonging to c2...
