I'm trying to get a better understanding of linear codes, so I decided to work on problems from various textbooks. I'm having trouble understanding how to do this problem, and I was wondering if anyone can lead me in the right direction.
Problem The matrix $G = [I_{4} | A]$, where $ G = \left[ \begin{array}{cccc|ccc} 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1\end{array} \right] $
is a genrator matrix in standard form for a $[7,4]$ binary code, denoted by $\mathcal{H}_3$. The parity check matrix for $\mathcal{H}_3$ is $H = [A^{T} | I_{3}] = \left[ \begin{array}{cccc|ccc} 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \end{array} \right]. $
Find at least four information sets in $\mathcal{H}_3$. Find at least one set of four coordinates that do not form an information set. $\blacksquare$
The book defines an information set as follows: Given a $[n,k]$ linear code $\mathcal{C}$, a generator matrix for $\mathcal{C}$ is any $k \times n$ matrix $G$ whose rows form a basis for $\mathcal{C}$. For any set of $k$ independent columns of $G$, the corresponding set of coordinates forms an information set for $\mathcal{C}$.
Any help would be greatly appreciated since I've been staring at this for quite some time. Thanks!