I got stuck in the following problem.
We recursively define the matrices for all $i\in \mathbb N$
$G_1 :=\begin{bmatrix} 1 &1\\ 0&1 \end{bmatrix}$
and $$G_{i+1} :=\begin{bmatrix} G_i &G_i \\ 0&J\\ \end{bmatrix}$$
for all $i\in \mathbb N$.
(a) Determine explicitly $G_2$ and $G_3$ and bring both to reduced row echelon form.
i. What (known) code does $G_2$ describe?
ii. Give a control matrix for $R_2$.
iii. $R_3$ is equivalent to the $(8, 4)$ Bauer code.
(b) Prove by induction:
$R_i$ contains the words 0 and 1. All other code words have the weight 2i-1.