MinT - Best Known (9, 9+9, s)-Nets in Base 2

Best Known (9, 9+9, s)-Nets in Base 2

(9, 9+9, 20)-Net over F₂ — Constructive and digital

Digital (9, 18, 20)-net over F₂, using

linear code embedding found in a computer search [i]

(9, 9+9, 27)-Net over F₂ — Upper bound on s (digital)

There is no digital (9, 18, 28)-net over F₂, because

1 times m-reduction [i] would yield digital (9, 17, 28)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷, 28, F₂, 8) (dual of [28, 11, 9]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁸, 29, F₂, 9) (dual of [29, 11, 10]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(9, 9+9, 34)-Net in Base 2 — Upper bound on s

There is no (9, 18, 35)-net in base 2, because

extracting embedded OOA [i] would yield OOA(2¹⁸, 35, S₂, 2, 9), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 4 415953 315895 784996 821323 246350 813671 372880 963446 730828 816384 / 16 236389 973973 339032 691828 244417 017864 713945 084685 578181 > 2¹⁸ [i]