Let $A_1$, $A_2$ be two linear codes over $F_q$ with parameters $[n_1, k_1, d_1]$ and $[n_2, k_2, d_2]$.
Let $A:= \{ (a_1 || a_2) | a_1 \in A_1, a_2 \in A_2\}$, where || is the sign for concatenation.
I read that A is a linear code and has distance $d = min\{d_1, d_2\}$. But why?
Hints :
To show that A is a linear subspace, you want to show that $(a_1||a_2) \oplus (a_3||a_4) \in A$ when $(a_1||a_2) \in A$ and $(a_3||a_4) \in A$. How can you rearrange that sum?
To show that $d = \min(d_1, d_2)$ it suffices to show that $d \le \min(d_1, d_2)$ (which can be done constructively by giving an example) and $d \ge \min(d_1, d_2)$ (which can be done by contradiction).