MinT - Best Known (156−77, 156, s)-Nets in Base 3

Best Known (156−77, 156, s)-Nets in Base 3

(156−77, 156, 64)-Net over F₃ — Constructive and digital

Digital (79, 156, 64)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 53, 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 (26, 103, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(156−77, 156, 84)-Net over F₃ — Digital

Digital (79, 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]

(156−77, 156, 626)-Net in Base 3 — Upper bound on s

There is no (79, 156, 627)-net in base 3, because

1 times m-reduction [i] would yield (79, 155, 627)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 89 972561 968480 214823 946264 803135 782023 497014 247285 537647 387621 084432 306877 > 3¹⁵⁵ [i]