MinT - Best Known (184−105, 184, s)-Nets in Base 3

Best Known (184−105, 184, s)-Nets in Base 3

(184−105, 184, 54)-Net over F₃ — Constructive and digital

Digital (79, 184, 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]

(184−105, 184, 84)-Net over F₃ — Digital

Digital (79, 184, 84)-net over F₃, using

t-expansion [i] based on digital (71, 184, 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]

(184−105, 184, 433)-Net in Base 3 — Upper bound on s

There is no (79, 184, 434)-net in base 3, because

1 times m-reduction [i] would yield (79, 183, 434)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2127 934978 819764 104468 645089 592673 916346 508624 421952 073194 560196 930560 122603 318476 893449 > 3¹⁸³ [i]