Assume that $q$ is odd. Let $C_{i}$ be an $\left[n,k_{i},d_{i}\right]-$linear code over $\mathbb{F}_{q}$, for $i=1,2$. Define $$C_{1}\diamondsuit C_{2}=\left\{ \left(c_{1}+c_{2},c_{1}-c_{2}\right):c_{1}\in C_{1},c_{2}\in C_{2}\right\}.$$ Let $d$ be the minimum distance of $C_{1}\diamondsuit C_{2}$. Show that $d=2d_{2}$ if $2d_{2}\le d_{1}$ and $d_{1}\leq d\leq2d_{2}$ if $2d_{2}>d_{1}$.
I have found that $$d=\min\left\{ 2d_{1},2d_{2},\max\left\{ d_{1},d_{2}\right\} \right\}. $$ And i got that $d=2d_{2}$ if $2d_{2}\le d_{1}$. I have a problem with the next condition, $d_{1}\leq d\leq2d_{2}$ if $2d_{2}>d_{1}$. I cant seem to get the right inequality.