MinT - Best Known (228−117, 228, s)-Nets in Base 3

Best Known (228−117, 228, s)-Nets in Base 3

(228−117, 228, 74)-Net over F₃ — Constructive and digital

Digital (111, 228, 74)-net over F₃, using

t-expansion [i] based on digital (107, 228, 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]

(228−117, 228, 104)-Net over F₃ — Digital

Digital (111, 228, 104)-net over F₃, using

t-expansion [i] based on digital (102, 228, 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]

(228−117, 228, 771)-Net in Base 3 — Upper bound on s

There is no (111, 228, 772)-net in base 3, because

1 times m-reduction [i] would yield (111, 227, 772)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 143769 487740 719470 049877 694086 912094 978552 264644 404674 583033 125509 495857 497681 073805 704680 592561 838665 091769 > 3²²⁷ [i]