Invertible binary matrix with maximum number of $2$-by-$2$ invertible submatrices

The problem is related to all-or-nothing transforms (AONT) suggested by Rivest as a preprocessing for encrypting data with a block cipher. Since then there have been various applications of AONTs in cryptography and security. D'Arco, Esfahani and Stinson posed the problem on the constructions of binary matrices for which the desired properties of an AONT hold with the maximum probability. That is, for given integers $t\le s$, what is the maximum number of $t$-by-$t$ invertible submatrices in a binary matrix of order $s$? For the case $t=2$, let $R_2(s)$ denote the maximal proportion of 2-by-2 invertible submatrices. D'Arco, Esfahani and Stinson conjectured that the limit is between 0.492 and 0.625. We completely solve the case $t=2$ by showing that $\lim_{s\rightarrow\infty}R_2(s)=0.5$.

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