MinT - Best Known (68−33, 68, s)-Nets in Base 3

Best Known (68−33, 68, s)-Nets in Base 3

(68−33, 68, 40)-Net over F₃ — Constructive and digital

Digital (35, 68, 40)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 20, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 48, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(68−33, 68, 47)-Net over F₃ — Digital

Digital (35, 68, 47)-net over F₃, using

net from sequence [i] based on digital (35, 46)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 35 and N(F) ≥ 47, using
  - a curve from the manYPoints database [i]

(68−33, 68, 323)-Net in Base 3 — Upper bound on s

There is no (35, 68, 324)-net in base 3, because

1 times m-reduction [i] would yield (35, 67, 324)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 96 242340 120620 274960 600320 005249 > 3⁶⁷ [i]