There is a matrix A where each entry is either 0 or 1. Each column has exactly a 1's and each row has at most b 1's. What's the upper bound of abs(|A|)? 
The condition is stronger than the Hadamard's maximal determinant problem. Is there any known result?



Estimate the determinant of sparse 0-1 matrix
It looks like a previous answer disappeared, so I will expand on my comments.


Using that a determinant corresponds to the volume of a parallelipiped, an immediate
upper bound (using columns) of $a^{n/2}$ results. If you know the distribution across the
rows, use the product of their lengths (don't forget the square root) for a slightly better bound.


There also has been work done on determinants of (I think binary but am not sure) matrices with constant row sums. I do not have any names or dates come to mind, but I suspect the work has been 
cited here on MO. You might succeed with a web search for "constant row sum".


