Hello,

I'm at a loss when it comes to the next step here

I've got the question:

Let C be the [6; 3] binary code with generator matrix

G =

1 1 0 0 0 0

0 1 1 0 0 0

1 1 1 1 1 1

Is C self-orthogonal? Do the codewords whose weights are divisible by

four form a subcode of C? Justify your answers.

the generator for the dual code, is the parity matrix for the code C

and a code C is self-orthgonal if $\displaystyle C \subseteq C^\perp$

and if $\displaystyle G=[I_k|A]$ is a generator matrix for the [n,k] code C in standard form then $\displaystyle H=[-A^T|I_{n-k}]$ is a parity check matrix for C

from this I took the matrix and got

G=

1 0 0 1 1 1

0 1 0 1 1 1

0 0 1 1 1 1

by row operations

then I get

$\displaystyle H=C^\perp =$

1 1 1 1 0 0

1 1 1 0 1 0

1 1 1 0 0 1

I don't know what to do from here to show $\displaystyle C \subseteq C^\perp$ or that it's not the case

I'm also unsure of what to do about the second part of the question, where it asks if the codewords of weight divisible by 4 form a subcode

edit:

ok so I know it's not the case

C is not self orthogonal, and the codewords of weight divisible by 4 are not a subcode of C

how do I prove this?