Consider $A$ is an $k\times n-k$ matrix over $\mathbb{F}_q$. Consider an $k \times n$ matrix $G=(I_k\mid A)$ as a generator of a code $C$.
My question: If $G\cdot G^T={\bf 0}$ over $\mathbb{F}_q$, then is it correct to say $G$ is not only generator of $C$, but also is parity check matrix of $C$? Is there a special name for the code $C$ in coding theory?
Thanks
Each codeword $c\in C$ has the form $c = u G$ for some vector $u\in {\Bbb F}_q^k$. Since $G G^t=0$, $cG^t = (uG)G^t = u(GG^t)=u0 = 0$ and so a dimension argument shows that $G^t$ is a a parity check matrix of $G$. If so, the code is called self-dual.