MinT - Best Known (228−143, 228, s)-Nets in Base 3

Best Known (228−143, 228, s)-Nets in Base 3

(228−143, 228, 60)-Net over F₃ — Constructive and digital

Digital (85, 228, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-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]

(228−143, 228, 84)-Net over F₃ — Digital

Digital (85, 228, 84)-net over F₃, using

t-expansion [i] based on digital (71, 228, 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]

(228−143, 228, 390)-Net in Base 3 — Upper bound on s

There is no (85, 228, 391)-net in base 3, because

1 times m-reduction [i] would yield (85, 227, 391)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 085547 019432 668010 020896 063073 286787 291029 916471 347751 640951 794008 277466 399106 678434 793786 905267 332900 465171 > 3²²⁷ [i]