MinT - Best Known (150−73, 150, s)-Nets in Base 3

Best Known (150−73, 150, s)-Nets in Base 3

(150−73, 150, 64)-Net over F₃ — Constructive and digital

Digital (77, 150, 64)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 51, 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, 99, 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]

(150−73, 150, 85)-Net over F₃ — Digital

Digital (77, 150, 85)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(150−73, 150, 638)-Net in Base 3 — Upper bound on s

There is no (77, 150, 639)-net in base 3, because

1 times m-reduction [i] would yield (77, 149, 639)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 123678 165702 839154 702235 212944 715423 881363 129281 180415 969603 670790 942649 > 3¹⁴⁹ [i]