With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)$ attending the Friedman bound $N\ge 2^n(1-n/2(t+1))$ are OA$(2048,14,2,7)$. Such array can be obtained by puncturing any binary $1$-perfect code of length $15$. We construct orthogonal arrays with these and similar parameters OA$(N=2^{n-m+1},n=2^m-2,2,t=2^{m-1}-1)$, $m\ge 4$, that are not punctured $1$-perfect codes. Additionally, we prove that any orthogonal array OA$(N,n,2,t)$ with even $t$ attending the bound $N \ge 2^n(1-(n+1)/2(t+2))$ induces an equitable $3$-partition of the $n$-cube.