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$ 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?

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  • $\begingroup$ Have you tried a few examples? Try to form a conjecture. $\endgroup$ – Yuval Filmus Oct 2 '14 at 12:27
  • $\begingroup$ 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. $\endgroup$ – ysb.4 Oct 2 '14 at 12:32

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