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

Denis S. Krotov

doi:10.1016/j.disc.2019.111659

paper

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

arXiv:1905.11371 · doi:10.1016/j.disc.2019.111659

Abstract

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 $N \geq 2^{n} (1 - n /2 (t + 1))$ 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 $(N = 2^{n - m + 1}, n = 2^{m} - 2, 2, t = 2^{m - 1} - 1)$ , $m \geq 4$ , 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 $N \geq 2^{n} (1 - (n + 1) /2 (t + 2))$ induces an equitable $3$ -partition of the $n$ -cube.

arXiv comment18pp. V.2: revised accepted version; the title was changed

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

Abstract

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