MinT - Best Known (67, 67+55, s)-Nets in Base 3

Best Known (67, 67+55, s)-Nets in Base 3

(67, 67+55, 68)-Net over F₃ — Constructive and digital

Digital (67, 122, 68)-net over F₃, using

trace code for nets [i] based on digital (6, 61, 34)-net over F₉, using
- net from sequence [i] based on digital (6, 33)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
    - a function field by SÃ©mirat [i]

(67, 67+55, 88)-Net over F₃ — Digital

Digital (67, 122, 88)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+55, 724)-Net in Base 3 — Upper bound on s

There is no (67, 122, 725)-net in base 3, because

1 times m-reduction [i] would yield (67, 121, 725)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5431 587254 638617 522639 305972 298663 931220 487530 000244 954379 > 3¹²¹ [i]