MinT - Best Known (67, 67+65, s)-Nets in Base 3

Best Known (67, 67+65, s)-Nets in Base 3

(67, 67+65, 56)-Net over F₃ — Constructive and digital

Digital (67, 132, 56)-net over F₃, using

9 times m-reduction [i] based on digital (67, 141, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 52, 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, 89, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(67, 67+65, 74)-Net over F₃ — Digital

Digital (67, 132, 74)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+65, 543)-Net in Base 3 — Upper bound on s

There is no (67, 132, 544)-net in base 3, because

1 times m-reduction [i] would yield (67, 131, 544)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 329 098311 055471 410410 701262 293473 560781 191995 233380 366711 803905 > 3¹³¹ [i]