Optimal Labeling Schemes for $Γ$-Free Matrices

We study labeling schemes for $p \times q$ $Γ$-free binary matrices, which do not contain a submatrix of the form $Γ= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$. We give a labeling scheme that uses $O(\log^2 N)$-bit labels for rows and columns, where $N=p+q$, such that any matrix entry can be det