MinT - Best Known (212−131, 212, s)-Nets in Base 3

Best Known (212−131, 212, s)-Nets in Base 3

(212−131, 212, 56)-Net over F₃ — Constructive and digital

Digital (81, 212, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-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]

(212−131, 212, 84)-Net over F₃ — Digital

Digital (81, 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]

(212−131, 212, 382)-Net in Base 3 — Upper bound on s

There is no (81, 212, 383)-net in base 3, because

1 times m-reduction [i] would yield (81, 211, 383)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 52832 552815 063033 275227 963546 527258 943897 419477 868683 521489 112412 420927 946253 102789 478618 248265 073279 > 3²¹¹ [i]