What is a linear binary code? What is the weight $w(C)$ of a linear binary code $C ?$ Define the bar product $C_{1} \mid C_{2}$ of two binary linear codes $C_{1}$ and $C_{2}$, stating the conditions that $C_{1}$ and $C_{2}$ must satisfy. Under these conditions show that

$$w\left(C_{1} \mid C_{2}\right) \geqslant \min \left(2 w\left(C_{1}\right), w\left(C_{2}\right)\right)$$