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Consider a v v (0; 1) matrix A with exactly k ones in each row and each column. A is (n){stable, if it does not contain any n submatrix with exactly one 0. If A is (n){stable, n 2, then under suitable conditions on A, v k n k(n-1)+(-2) -2. The case of equality leads to new and substantive connections with block designs. The previous bound and characterization of (2){stable matrices follows immediately as a special case. [ABSTRACT FROM AUTHOR]