MinT - Best Known (135−69, 135, s)-Nets in Base 3

Best Known (135−69, 135, s)-Nets in Base 3

(135−69, 135, 56)-Net over F₃ — Constructive and digital

Digital (66, 135, 56)-net over F₃, using

3 times m-reduction [i] based on digital (66, 138, 56)-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 (15, 87, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(135−69, 135, 67)-Net over F₃ — Digital

Digital (66, 135, 67)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−69, 135, 481)-Net in Base 3 — Upper bound on s

There is no (66, 135, 482)-net in base 3, because

1 times m-reduction [i] would yield (66, 134, 482)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9024 520426 103635 441560 374035 175615 004937 004615 006651 271191 278485 > 3¹³⁴ [i]