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?

Question

$ 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?