Let B be a square matrix, let I be identity matrix of the same size, and let G be the generator matrix in standard form created by appending B to I. Prove that the code over GF(2) generated by G is self-dual. I'm stuck on how to get the relationship that B*B^T=I. B^T means the transpose of B.
2026-03-27 16:53:01.1774630381
Matrix over GF(2)
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As posed, the statement is not always true. Consider for example the following generator matrix G:
$$G=\begin{bmatrix} 1&0&1&1\\0&1&0&0 \end{bmatrix}$$
It answers all the requirements, yet $BB^t\neq I$ and the code generated by this matrix obviously isn't self-dual (easy to check - the first codeword in $G$ times itself isn't $0$).
Perhaps there are further details regarding $B$ or the code? Perhaps $B$ is given?