MinT - Best Known (188−109, 188, s)-Nets in Base 3

Best Known (188−109, 188, s)-Nets in Base 3

(188−109, 188, 54)-Net over F₃ — Constructive and digital

Digital (79, 188, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-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]

(188−109, 188, 84)-Net over F₃ — Digital

Digital (79, 188, 84)-net over F₃, using

t-expansion [i] based on digital (71, 188, 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]

(188−109, 188, 419)-Net in Base 3 — Upper bound on s

There is no (79, 188, 420)-net in base 3, because

1 times m-reduction [i] would yield (79, 187, 420)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 174243 294558 015087 471453 455158 170841 419938 446546 392101 751033 388711 148430 252541 446916 509081 > 3¹⁸⁷ [i]