MinT - Best Known (235−123, 235, s)-Nets in Base 3

Best Known (235−123, 235, s)-Nets in Base 3

(235−123, 235, 74)-Net over F₃ — Constructive and digital

Digital (112, 235, 74)-net over F₃, using

t-expansion [i] based on digital (107, 235, 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]

(235−123, 235, 104)-Net over F₃ — Digital

Digital (112, 235, 104)-net over F₃, using

t-expansion [i] based on digital (102, 235, 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]

(235−123, 235, 738)-Net in Base 3 — Upper bound on s

There is no (112, 235, 739)-net in base 3, because

1 times m-reduction [i] would yield (112, 234, 739)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4659 546940 400793 421913 024258 262286 269246 160865 307002 813029 014015 328117 433442 566374 965154 291953 223868 761052 396943 > 3²³⁴ [i]