So here is the question and I am trying to find a proof for the minimum hamming distance d3:
Let $C_1$ be a $(n_1, M, d_1)$ q-ary code and $C_2$ a $(n_2, M, d_2)$ q-ary code. Assume that we have named the codewords:
$$C_1 = {w_1, w_2, . . . , w_M }$$ $$C_2 = {v_1, v_2, . . . , v_M }$$ Define the new code $C_3$ as the set of words $wivi$ for $i = 1, . . . , M$ . Find with justification the parameters $n$, $M$, $d$ for the code $C_3$. (The data may not be enough to determine the distance d of the new code C3 precisely. But you can say that d is greater than some number. Which number?)
This is what I have:
Assume $$w_i, w_j \in C1$$ s.t $$d(wi, wj) = d1$$ Then, $$d(wivi, wjvj) = d1 + d(vi, vj)$$, so $$d(wivi, wjvj) >= d1 + 1$$ Assume $$vk, vl \in C2$$ s.t $$d(vk, vl) = d2$$ Then, $$d(wkvk, wlvl) = d2 + d(vk, vl)$$, so $$d(wkvk, wlvl) >= d2 + 1$$ By above, $$d3 >= max\{d1, d2\} + 1$$
Thank you