I need to construct an $m \times n$ binary matrix $B$ from a given row and columns distribution.
What algorithms can be used for this? As a concrete example : $B$ a $12 \times 63$ matrix with row distribution $12 r^{19}$ and column distribution $12 c^2 + 39 c^3 + 9 c^7 + 3 c^8$. That is
all $12$ rows have weight $19$; $12$ columns have weight $2$, $39$ columns have weight $3$, $\cdots$, $3$ columns have weight $8$. The weight of binary row or column is its sum or equivalently the number of nonzero entries. (This comes up in LDPC code constructions; the example is from [LDPC Code Designs (2017)](https://dl.acm.org/doi/10.5555/3137384), page 38)