Geodesic
The maximum number of $2\times 2$ odd submatrices in $(0,1)$-matrices
The electronic journal of linear algebra, Tome 10 (2003), pp. 223-231.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let A be an m * n, (0, 1)-matrix. A submatrix of A is odd if the sum of its entries is an odd integer and even otherwise. The maximum number of 2*2 odd submatrices in a (0, 1)-matrix is related to the existence of Hadamard matrices and bounds on Tur'an numbers. Pinelis [On the minimal number of even submatrices of 0-1 matrices, Designs, Codes and Cryptography, 9:85-93, 1994] exhibits an asymptotic formula for the minimum possible number of p * q even submatrices of an m * n (0, 1)-matrix. Assuming the Hadamard conjecture, specific techniques are provided on how to assign the 0's and 1's, in order to yield the maximum number of 2 * 2 odd submatrices in an m * n (0, 1)-matrix. Moreover, formulas are determined that yield the exact maximum counts with one exception, in which case upper and lower bounds are given. These results extend and refine those of Pinelis.
Classification : 15A36, 15A37
Keywords: (0, 1)-matrices, even and odd matrices, Hadamard matrices 
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     title = {The maximum number of $2\times 2$ odd submatrices in $(0,1)$-matrices},
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Marks, Michael; Norwood, Rick; Poole, George. The maximum number of $2\times 2$ odd submatrices in $(0,1)$-matrices. The electronic journal of linear algebra, Tome 10 (2003), pp. 223-231. http://geodesic.mathdoc.fr/item/ELA_2003__10__a7/