In coding theory, there are parity-check codes whose parity-check matrices $H$ are generated via column permutations. For instance, the binary LDPC codes constructed in Gallager's 1962 IRE Trans paper uses the following $H$ matrix:

$$H = \left[\begin{array}{c} X_1\\ X_2\\ \vdots\\ X_n \end{array}\right]$$

where submatrices $X_i$, $2 \leq i \leq n$ are obtained by randomly permuting columns of $X_1$ of certain kind. However, to make the codes suitable to iterative decoding, typically we impose one restriction which requires that any two row vectors in $H$ mustn't have 2 or more overlapping nonzero elements. In other words, we would like $H$ to be free of $2 \times 2$ all-one matrix.

I tried to write a program to do that, but so far my effort is not good. I'm wondering if there is any known algorithmic way to adjust the permutated submatrices $X_1, \dots, X_n$ so that the overlapping constraint is satisfied?

Thanks for your help!