MinT - Best Known (117−57, 117, s)-Nets in Base 3

Best Known (117−57, 117, s)-Nets in Base 3

(117−57, 117, 56)-Net over F₃ — Constructive and digital

Digital (60, 117, 56)-net over F₃, using

3 times m-reduction [i] based on digital (60, 120, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 45, 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, 75, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(117−57, 117, 69)-Net over F₃ — Digital

Digital (60, 117, 69)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(117−57, 117, 508)-Net in Base 3 — Upper bound on s

There is no (60, 117, 509)-net in base 3, because

1 times m-reduction [i] would yield (60, 116, 509)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 889918 881551 252138 697867 557040 978879 610103 629599 431345 > 3¹¹⁶ [i]