MinT - Best Known (139−55, 139, s)-Nets in Base 3

Best Known (139−55, 139, s)-Nets in Base 3

(139−55, 139, 128)-Net over F₃ — Constructive and digital

Digital (84, 139, 128)-net over F₃, using

3 times m-reduction [i] based on digital (84, 142, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 71, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(139−55, 139, 142)-Net over F₃ — Digital

Digital (84, 139, 142)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−55, 139, 1473)-Net in Base 3 — Upper bound on s

There is no (84, 139, 1474)-net in base 3, because

1 times m-reduction [i] would yield (84, 138, 1474)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 704181 755399 085186 584702 939069 411072 462185 684897 939540 859946 686433 > 3¹³⁸ [i]