MinT - Best Known (227−117, 227, s)-Nets in Base 3

Best Known (227−117, 227, s)-Nets in Base 3

(227−117, 227, 74)-Net over F₃ — Constructive and digital

Digital (110, 227, 74)-net over F₃, using

t-expansion [i] based on digital (107, 227, 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]

(227−117, 227, 104)-Net over F₃ — Digital

Digital (110, 227, 104)-net over F₃, using

t-expansion [i] based on digital (102, 227, 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]

(227−117, 227, 755)-Net in Base 3 — Upper bound on s

There is no (110, 227, 756)-net in base 3, because

1 times m-reduction [i] would yield (110, 226, 756)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 689700 899601 114702 583088 277264 048795 282789 385264 270357 860040 256052 806925 131556 359239 517023 225592 264792 336985 > 3²²⁶ [i]