MinT - Best Known (159, 238, s)-Nets in Base 3

Best Known (159, 238, s)-Nets in Base 3

(159, 238, 164)-Net over F₃ — Constructive and digital

Digital (159, 238, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 46, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (113, 192, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 96, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(159, 238, 375)-Net over F₃ — Digital

Digital (159, 238, 375)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 238, 6068)-Net in Base 3 — Upper bound on s

There is no (159, 238, 6069)-net in base 3, because

1 times m-reduction [i] would yield (159, 237, 6069)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 119944 462767 829821 308774 823274 335328 803932 495318 118004 787577 497739 564347 068594 987336 028854 875507 801361 001991 456027 > 3²³⁷ [i]