MinT - Best Known (156−81, 156, s)-Nets in Base 3

Best Known (156−81, 156, s)-Nets in Base 3

(156−81, 156, 56)-Net over F₃ — Constructive and digital

Digital (75, 156, 56)-net over F₃, using

9 times m-reduction [i] based on digital (75, 165, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 60, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 105, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(156−81, 156, 84)-Net over F₃ — Digital

Digital (75, 156, 84)-net over F₃, using

t-expansion [i] based on digital (71, 156, 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]

(156−81, 156, 518)-Net in Base 3 — Upper bound on s

There is no (75, 156, 519)-net in base 3, because

1 times m-reduction [i] would yield (75, 155, 519)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 94 508490 915071 085509 731257 270583 133379 023125 179783 067550 970483 693805 619505 > 3¹⁵⁵ [i]