I have to solve a problem on Linear Code, I've studied the theory but since I've never saw nor done an exercise on it I'm pretty stuck with lot of different definitions and method and I'm not sure how to solve this problem.
Is given a linear code C, binary, generated from the set of codeword {1011,1101}.Question:
- Write the control matrix
- Write all the lateral class of $F_4^2/C$
- Write all the leaders, in case they're multiple choose one
- Write the table of leaders and all their sydromes
Decode the word 1110 1111 0011
I wrote the generatrix matrix just using the 2 codeword at the beginning, then I modified the matrix with linear operations to get a standard form.And using that matrix I got the Control Matrix which is at the end\begin{matrix} 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ \end{matrix}
But now I don't know how to continue. Hope someone can help me.
Update1
I have $F_4^2$:
0000 1000 0100 0010 0001 1100 0110 0011 1010 0101 1001 0111 1110 1101 1011 1111
and C:{1011,0000,1101,0110}
so $F_4^2/C$: {0000,1000,0100,0010,0001,0110,1011,1101}
And I have to find the coset of each of these elements
is it correct?
Update 2:
$[0000]=(0000; 1011; 1101; 0110) \\
[1000] = (1000; 0011; 0101; 1110) \\
[0100] = (0100; 1111; 1001; 0010) \\
[0001] = (0001; 1010; 1100; 0111)$
The representant of the coset is the leader.
At the end I think I can answer my own question summing up all the things I've done. The check matrix is written over there and it's correct. The cosets are: $[0000]=(0000; 1011; 1101; 0110) \\ [1000] = (1000; 0011; 0101; 1110) \\ [0100] = (0100; 1111; 1001; 0010) \\ [0001] = (0001; 1010; 1100; 0111)$
aswering the point b)
The representant of the coset is the leader.In the third coset I had 2 members with the same weights so I chose one. answering the point c)
The sydromes are respectively:
$[0 0];[1 1];[1 0];[0 1];$ answering the point d)
Decoding the word $[1110]$: $H*w^t=[11]$ it is in the coset of leader [1000] so $c=(1110)+(1000)=(0110)$
In the same way we can decode the other words which are respectively: $(1011); (1011) $
answering the point e)