MinT - Best Known (245−135, 245, s)-Nets in Base 3

Best Known (245−135, 245, s)-Nets in Base 3

(245−135, 245, 74)-Net over F₃ — Constructive and digital

Digital (110, 245, 74)-net over F₃, using

t-expansion [i] based on digital (107, 245, 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]

(245−135, 245, 104)-Net over F₃ — Digital

Digital (110, 245, 104)-net over F₃, using

t-expansion [i] based on digital (102, 245, 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]

(245−135, 245, 640)-Net in Base 3 — Upper bound on s

There is no (110, 245, 641)-net in base 3, because

1 times m-reduction [i] would yield (110, 244, 641)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 273 117051 505306 038814 154438 284915 056414 583375 691138 818241 560931 575903 533112 739028 775371 094154 481358 813685 429724 849435 > 3²⁴⁴ [i]