Apologize if you've read my question on Mathoverflow, I'm very curious about whether there's an answer to this.
In coding theory, there are parity-check codes whose parity-check matrices H are generated via column permutations. For instance, the LDPC codes constructed in Gallager's 1962 IRE Trans paper uses the following $H$ matrix: $ H = \begin{bmatrix} \text{---} &X_1 &\text{---}\\ \text{---} &X_2 &\text{---}\\ &\vdots &\\ \text{---} &X_n &\text{---}\\ \end{bmatrix} $ where submatrices $X_2,\ldots,X_n$ are just random column permutations of $X_1$. However, to make the codes efficient in decoding, there is one restriction which requires that any two row vectors in $H$ mustn't have 2 or more overlapping elements. By overlapping, I mean for two different row vectors of $H$, say $V_a$ and $V_b$, there exists an index $i$ s.t. $V_a[i] = V_b[i]$;
I tried to write a program to do that, but so far my effort is not good.
The question is: Is there any known algorithmic way to adjust the permutated submatrices $X_1,\ldots,X_n$ so that the overlapping constraint is satisfied?