MinT - Best Known (216−123, 216, s)-Nets in Base 3

Best Known (216−123, 216, s)-Nets in Base 3

(216−123, 216, 64)-Net over F₃ — Constructive and digital

Digital (93, 216, 64)-net over F₃, using

t-expansion [i] based on digital (89, 216, 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]

(216−123, 216, 96)-Net over F₃ — Digital

Digital (93, 216, 96)-net over F₃, using

t-expansion [i] based on digital (89, 216, 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]

(216−123, 216, 508)-Net in Base 3 — Upper bound on s

There is no (93, 216, 509)-net in base 3, because

1 times m-reduction [i] would yield (93, 215, 509)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 206506 916312 389238 737525 545647 945199 456060 396505 618402 769467 120725 147733 538271 610852 020471 840534 721451 > 3²¹⁵ [i]