MinT - Best Known (228−127, 228, s)-Nets in Base 3

Best Known (228−127, 228, s)-Nets in Base 3

(228−127, 228, 68)-Net over F₃ — Constructive and digital

Digital (101, 228, 68)-net over F₃, using

net from sequence [i] based on digital (101, 67)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 67)-sequence over F₉, using
  - s-reduction 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]

(228−127, 228, 96)-Net over F₃ — Digital

Digital (101, 228, 96)-net over F₃, using

t-expansion [i] based on digital (89, 228, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(228−127, 228, 576)-Net in Base 3 — Upper bound on s

There is no (101, 228, 577)-net in base 3, because

1 times m-reduction [i] would yield (101, 227, 577)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 146174 687953 443780 571377 333720 132736 060409 656373 896862 347006 774424 793741 648432 184907 244564 114401 675341 230763 > 3²²⁷ [i]