Tensor product of linear codes and generators

Question

Suppose I have a $[n_1,k_1,d_1]$-linear code $C_1$ and a $[n_2,k_2,d_2]$-linear code $C_2$. I want to show that $C_1 \otimes C_2$ is a $[n_1 n_2, k_1 k_2, d_1 d_2]$-linear code. Suppose they have generator matrices $G_1$ and $G_2$. Is it true that $G_1\otimes G_2$ (as vector space elements) generates $C_1 \otimes C_2$? If not, how do I show this?

Answer 1

Yes, the direct product code is denoted in this way and has this generator matrix (see MacWilliams, Sloane p.569). Some authors define $C_1 \otimes C_2$ to be the code with parity check matrix $H_1 \otimes H_2$. To see that the generator matrix definition is equivalent, $$(G_1 \otimes G_2)(H_1 \otimes H_2)^T = (G_1 \otimes G_2)(H_1^T \otimes H_2^T) = G_1 H_1^T \otimes G_2 H_2^T = 0 \otimes 0 = 0.$$ Likewise, $(H_1 \otimes H_2)(G_1 \otimes G_2)^T = 0$.