MinT - Best Known (2, 2+7, s)-Nets in Base 3

Best Known (2, 2+7, s)-Nets in Base 3

(2, 2+7, 9)-Net over F₃ — Constructive and digital

Digital (2, 9, 9)-net over F₃, using

a shift-net [i]

(2, 2+7, 11)-Net over F₃ — Upper bound on s (digital)

There is no digital (2, 9, 12)-net over F₃, because

1 times m-reduction [i] would yield digital (2, 8, 12)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
  - residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
    - improvement by Maruta on the Griesmer bound [i]

(2, 2+7, 13)-Net in Base 3 — Upper bound on s

There is no (2, 9, 14)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁹, 14, S₃, 7), but
- the linear programming bound shows that MÂ â‰¥ 885735 / 44 > 3⁹ [i]