MinT - Best Known (2, 8, s)-Nets in Base 3

Best Known (2, 8, s)-Nets in Base 3

(2, 8, 9)-Net over F₃ — Constructive and digital

Digital (2, 8, 9)-net over F₃, using

1 times m-reduction [i] based on digital (2, 9, 9)-net over F₃, using
- a shift-net [i]

(2, 8, 11)-Net over F₃ — Upper bound on s (digital)

There is no digital (2, 8, 12)-net over F₃, because

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, 8, 14)-Net in Base 3 — Upper bound on s

There is no (2, 8, 15)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6971 > 3⁸ [i]