MinT - Best Known (212−129, 212, s)-Nets in Base 3

Best Known (212−129, 212, s)-Nets in Base 3

(212−129, 212, 58)-Net over F₃ — Constructive and digital

Digital (83, 212, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(212−129, 212, 84)-Net over F₃ — Digital

Digital (83, 212, 84)-net over F₃, using

t-expansion [i] based on digital (71, 212, 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]

(212−129, 212, 401)-Net in Base 3 — Upper bound on s

There is no (83, 212, 402)-net in base 3, because

1 times m-reduction [i] would yield (83, 211, 402)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 50888 829546 827547 105606 752210 267752 073156 790166 845611 510256 362794 293805 943742 886170 666236 221181 622913 > 3²¹¹ [i]