MinT - Best Known (125−61, 125, s)-Nets in Base 3

Best Known (125−61, 125, s)-Nets in Base 3

(125−61, 125, 56)-Net over F₃ — Constructive and digital

Digital (64, 125, 56)-net over F₃, using

7 times m-reduction [i] based on digital (64, 132, 56)-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 (15, 83, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(125−61, 125, 72)-Net over F₃ — Digital

Digital (64, 125, 72)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−61, 125, 535)-Net in Base 3 — Upper bound on s

There is no (64, 125, 536)-net in base 3, because

1 times m-reduction [i] would yield (64, 124, 536)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 146485 376727 583794 444984 730619 459823 344286 272002 986057 867025 > 3¹²⁴ [i]