MinT - Best Known (69, 69+55, s)-Nets in Base 3

Best Known (69, 69+55, s)-Nets in Base 3

(69, 69+55, 72)-Net over F₃ — Constructive and digital

Digital (69, 124, 72)-net over F₃, using

trace code for nets [i] based on digital (7, 62, 36)-net over F₉, using
- net from sequence [i] based on digital (7, 35)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 7 and N(F) ≥ 36, using
    - a function field by SÃ©mirat [i]

(69, 69+55, 94)-Net over F₃ — Digital

Digital (69, 124, 94)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+55, 788)-Net in Base 3 — Upper bound on s

There is no (69, 124, 789)-net in base 3, because

1 times m-reduction [i] would yield (69, 123, 789)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49452 293995 295454 267484 583292 309763 000569 540910 652338 724875 > 3¹²³ [i]