MinT - Best Known (135−65, 135, s)-Nets in Base 3

Best Known (135−65, 135, s)-Nets in Base 3

(135−65, 135, 60)-Net over F₃ — Constructive and digital

Digital (70, 135, 60)-net over F₃, using

3 times m-reduction [i] based on digital (70, 138, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 49, 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, 89, 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]

(135−65, 135, 82)-Net over F₃ — Digital

Digital (70, 135, 82)-net over F₃, using

t-expansion [i] based on digital (69, 135, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(135−65, 135, 605)-Net in Base 3 — Upper bound on s

There is no (70, 135, 606)-net in base 3, because

1 times m-reduction [i] would yield (70, 134, 606)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8756 617767 598789 898261 009389 825351 124440 444306 168853 207670 787905 > 3¹³⁴ [i]