MinT - Best Known (219−117, 219, s)-Nets in Base 3

Best Known (219−117, 219, s)-Nets in Base 3

(219−117, 219, 69)-Net over F₃ — Constructive and digital

Digital (102, 219, 69)-net over F₃, using

net from sequence [i] based on digital (102, 68)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 68)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(219−117, 219, 104)-Net over F₃ — Digital

Digital (102, 219, 104)-net over F₃, using

net from sequence [i] based on digital (102, 103)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(219−117, 219, 641)-Net in Base 3 — Upper bound on s

There is no (102, 219, 642)-net in base 3, because

1 times m-reduction [i] would yield (102, 218, 642)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104 256592 040997 217720 146263 984500 807619 630163 646579 405882 756996 258625 733113 804759 794652 050945 467708 394821 > 3²¹⁸ [i]