Let the matrix A be the (nxk)-binary random matrix such that n>k.
And define the column-weight as the number of 1's in the column.
Then,, I want to know the uniform distribution of the column-weight can help the matrix A has the full-rank?
Simply, should all the columns of matrix A have the same number of 1's to achieve the full rank property?

Intuitively, it may be right... But I cannot prove this clearly.
I'll wait the helpful relply from somebody.