MinT - Best Known (188−99, 188, s)-Nets in Base 3

Best Known (188−99, 188, s)-Nets in Base 3

(188−99, 188, 64)-Net over F₃ — Constructive and digital

Digital (89, 188, 64)-net over F₃, using

net from sequence [i] based on digital (89, 63)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
    - a function field by GarcÃa and Quoos [i]

(188−99, 188, 96)-Net over F₃ — Digital

Digital (89, 188, 96)-net over F₃, using

net from sequence [i] based on digital (89, 95)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(188−99, 188, 585)-Net in Base 3 — Upper bound on s

There is no (89, 188, 586)-net in base 3, because

1 times m-reduction [i] would yield (89, 187, 586)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 172019 828186 874476 228340 349466 560330 221363 778097 669443 285053 048500 526705 736887 484781 024181 > 3¹⁸⁷ [i]