On the orthogonal arrays of parameters OA(1536,13,2,7) and related
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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) (or, in a different notation, OA_12(7,13,2)), 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 nonequivalent orthogonal arrays OA(768,12,2,6) (or OA_12(6,12,2)), 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< 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 an orthogonal array with these and similar parameters OA(N=2^n-m+1,n=2^m-2,2,t=2^m-1-1), m> 4, that is not a punctured 1-perfect code. Additionally, we prove that any orthogonal array OA(N,n,2,t) with even t attending the bound N > 2^n(1-(n+1)/2(t+2)) induces an equitable 3-partition of the n-cube.
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