MinT - Best Known (213−109, 213, s)-Nets in Base 3

Best Known (213−109, 213, s)-Nets in Base 3

(213−109, 213, 71)-Net over F₃ — Constructive and digital

Digital (104, 213, 71)-net over F₃, using

net from sequence [i] based on digital (104, 70)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 70)-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]

(213−109, 213, 104)-Net over F₃ — Digital

Digital (104, 213, 104)-net over F₃, using

t-expansion [i] based on digital (102, 213, 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]

(213−109, 213, 730)-Net in Base 3 — Upper bound on s

There is no (104, 213, 731)-net in base 3, because

1 times m-reduction [i] would yield (104, 212, 731)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 143340 416078 082021 513758 596435 338850 247984 870648 382757 120118 374190 216254 781435 555405 520102 010164 537261 > 3²¹² [i]