MinT - Best Known (139−59, 139, s)-Nets in Base 3

Best Known (139−59, 139, s)-Nets in Base 3

(139−59, 139, 80)-Net over F₃ — Constructive and digital

Digital (80, 139, 80)-net over F₃, using

5 times m-reduction [i] based on digital (80, 144, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 72, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(139−59, 139, 116)-Net over F₃ — Digital

Digital (80, 139, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−59, 139, 1059)-Net in Base 3 — Upper bound on s

There is no (80, 139, 1060)-net in base 3, because

1 times m-reduction [i] would yield (80, 138, 1060)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 704364 593965 979905 628827 692473 409762 648583 702418 360269 702431 588777 > 3¹³⁸ [i]