MinT - Best Known (242−133, 242, s)-Nets in Base 3

Best Known (242−133, 242, s)-Nets in Base 3

(242−133, 242, 74)-Net over F₃ — Constructive and digital

Digital (109, 242, 74)-net over F₃, using

t-expansion [i] based on digital (107, 242, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(242−133, 242, 104)-Net over F₃ — Digital

Digital (109, 242, 104)-net over F₃, using

t-expansion [i] based on digital (102, 242, 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]

(242−133, 242, 638)-Net in Base 3 — Upper bound on s

There is no (109, 242, 639)-net in base 3, because

1 times m-reduction [i] would yield (109, 241, 639)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 852030 014683 906103 127664 886283 044672 927326 145197 297064 514143 507958 511728 744401 140237 442180 616631 349049 966764 491901 > 3²⁴¹ [i]