# G1, G2, H1, H2 are submatrices of the generator and parity check matrices of a code as described below. Can G1 * H2' and G2 * H1' both be all zeroes?

$G$ and $H$ are the generator and parity check matrices respectively of a linear block code.

Let $G_1 = G(:, 1:n-s)$ (Matlab style representation of sub matrices). That is, $G_1$ is equal to the first $n-s$ columns of $G$, and $G_2 = G(:, n-s+1:n)$. Let $H_1 = H(:, 1:s)$ and $H_2 = H(:, s+1:n)$, where $s \in \{1, 2, ... n-1\}$; it represents the number of columns in $G_2$.

Is it possible for $G_1 H_2'$ and $G_2 H_1'$ both to be all zeroes?

• Have you tried a few examples? Try to form a conjecture. – Yuval Filmus Oct 2 '14 at 12:27
• I have simulated it for various values of n and k by generating random linear block codes. Haven't yet found a code for which both products are all zeroes. – ysb.4 Oct 2 '14 at 12:32