MinT - Best Known (56, 56+51, s)-Nets in Base 3

Best Known (56, 56+51, s)-Nets in Base 3

(56, 56+51, 56)-Net over F₃ — Constructive and digital

Digital (56, 107, 56)-net over F₃, using

1 times m-reduction [i] based on digital (56, 108, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 41, 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, 67, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(56, 56+51, 68)-Net over F₃ — Digital

Digital (56, 107, 68)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+51, 512)-Net in Base 3 — Upper bound on s

There is no (56, 107, 513)-net in base 3, because

1 times m-reduction [i] would yield (56, 106, 513)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 384 904334 386812 622507 015549 805706 661793 107948 408483 > 3¹⁰⁶ [i]