MinT - Best Known (10−7, 10, s)-Nets in Base 3

Best Known (10−7, 10, s)-Nets in Base 3

(10−7, 10, 11)-Net over F₃ — Constructive and digital

Digital (3, 10, 11)-net over F₃, using

linear code embedding found in a computer search [i]

(10−7, 10, 15)-Net over F₃ — Upper bound on s (digital)

There is no digital (3, 10, 16)-net over F₃, because

1 times m-reduction [i] would yield digital (3, 9, 16)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁹, 16, F₃, 6) (dual of [16, 7, 7]-code), but
  - â€œvE2â€ bound on codes from Brouwerâ€™s database [i]

(10−7, 10, 21)-Net in Base 3 — Upper bound on s

There is no (3, 10, 22)-net in base 3, because

1 times m-reduction [i] would yield (3, 9, 22)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁹, 22, S₃, 2, 6), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 287963 682390 / 14 415719 > 3⁹ [i]