MinT - Best Known (128−59, 128, s)-Nets in Base 3

Best Known (128−59, 128, s)-Nets in Base 3

(128−59, 128, 64)-Net over F₃ — Constructive and digital

Digital (69, 128, 64)-net over F₃, using

trace code for nets [i] based on digital (5, 64, 32)-net over F₉, using
- net from sequence [i] based on digital (5, 31)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
    - a function field by GarcÃa and Quoos [i]

(128−59, 128, 86)-Net over F₃ — Digital

Digital (69, 128, 86)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−59, 128, 689)-Net in Base 3 — Upper bound on s

There is no (69, 128, 690)-net in base 3, because

1 times m-reduction [i] would yield (69, 127, 690)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 088804 784979 798028 202910 339255 086278 575458 247222 760462 990477 > 3¹²⁷ [i]