On the OA(1536,13,2,7) and related orthogonal arrays

Abstract

With a computer-aided approach based on connection with equitable partitions, we establish the uniqueness of the orthogonal array OA (1536,13,2,7), constructed in Fon-Der-Flaass (2017) 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≥2n(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=2n−m+1,n=2m−2,2,t=2m−1−1), m≥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≥2n(1−(n+1)∕2(t+2)) induces an equitable 3-partition of the n-cube.