MinT - Best Known (195−115, 195, s)-Nets in Base 3

Best Known (195−115, 195, s)-Nets in Base 3

(195−115, 195, 55)-Net over F₃ — Constructive and digital

Digital (80, 195, 55)-net over F₃, using

net from sequence [i] based on digital (80, 54)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 54)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(195−115, 195, 84)-Net over F₃ — Digital

Digital (80, 195, 84)-net over F₃, using

t-expansion [i] based on digital (71, 195, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(195−115, 195, 410)-Net in Base 3 — Upper bound on s

There is no (80, 195, 411)-net in base 3, because

1 times m-reduction [i] would yield (80, 194, 411)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 385 430791 302828 701723 660559 984704 075748 292712 703146 085640 592362 213168 905415 071173 350293 092103 > 3¹⁹⁴ [i]