**Comment by the author, April 6, 2000**: We can give
an improved lower bound for the modulo m rank of co-triangle matrices.

Theorem. Let A be an *n x n* co-triangle matrix over
a ring R, where *|R|=m*. Then *rank _{R}(A) >= log_{m}(n)*.

Proof. Let *A=BC* over *R*, where *B* is
an *n x r* and *C* is an *r x n* matrix. Since every column
of *A* is different, all columns of *C* should also be different.
The entries of *C* are chosen from the ring *R* of cardinality
*m*, so there are at most *m ^{r}* pairwise different
vectors of length