MinT - Best Known (66, 66+59, s)-Nets in Base 3

Best Known (66, 66+59, s)-Nets in Base 3

(66, 66+59, 60)-Net over F₃ — Constructive and digital

Digital (66, 125, 60)-net over F₃, using

1 times m-reduction [i] based on digital (66, 126, 60)-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 (21, 81, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]

(66, 66+59, 79)-Net over F₃ — Digital

Digital (66, 125, 79)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+59, 612)-Net in Base 3 — Upper bound on s

There is no (66, 125, 613)-net in base 3, because

1 times m-reduction [i] would yield (66, 124, 613)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 152013 662406 051714 009441 584701 924715 798049 670951 952617 714299 > 3¹²⁴ [i]