MinT - Best Known (126−63, 126, s)-Nets in Base 3

Best Known (126−63, 126, s)-Nets in Base 3

(126−63, 126, 56)-Net over F₃ — Constructive and digital

Digital (63, 126, 56)-net over F₃, using

3 times m-reduction [i] based on digital (63, 129, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 48, 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, 81, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(126−63, 126, 68)-Net over F₃ — Digital

Digital (63, 126, 68)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−63, 126, 491)-Net in Base 3 — Upper bound on s

There is no (63, 126, 492)-net in base 3, because

1 times m-reduction [i] would yield (63, 125, 492)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 456012 868856 632866 850047 351505 201563 110840 045031 826537 618065 > 3¹²⁵ [i]