The classification of orthogonal arrays OA(2048,14,2,7) and some completely regular codes

We describe the classification of orthogonal arrays OA$(2048,14,2,7)$, or, equivalently, completely regular $\{14;2\}$-codes in the $14$-cube ($30848$ equivalence classes). In particular, we find that there is exactly one almost-OA$(2048,14,2,7+1)$, up to equivalence. As derived objects, OA$(1024,13,2,6)$ ($202917$ classes) and completely regular $\{12,2;2,12\}$- and $\{14, 12, 2; 2, 12, 14\}$-codes in the $13$- and $14$-cubes, respectively, are also classified.