MinT - Best Known (82, 82+107, s)-Nets in Base 3

Best Known (82, 82+107, s)-Nets in Base 3

(82, 82+107, 57)-Net over F₃ — Constructive and digital

Digital (82, 189, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-sequence over F₉, using
  - s-reduction 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]

(82, 82+107, 84)-Net over F₃ — Digital

Digital (82, 189, 84)-net over F₃, using

t-expansion [i] based on digital (71, 189, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(82, 82+107, 456)-Net in Base 3 — Upper bound on s

There is no (82, 189, 457)-net in base 3, because

1 times m-reduction [i] would yield (82, 188, 457)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 507293 739447 229496 868712 188415 125559 125830 678604 703289 843294 879467 393962 143730 678952 094787 > 3¹⁸⁸ [i]