MinT - Best Known (128−53, 128, s)-Nets in Base 3

Best Known (128−53, 128, s)-Nets in Base 3

(128−53, 128, 80)-Net over F₃ — Constructive and digital

Digital (75, 128, 80)-net over F₃, using

6 times m-reduction [i] based on digital (75, 134, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 67, 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]

(128−53, 128, 117)-Net over F₃ — Digital

Digital (75, 128, 117)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−53, 128, 1104)-Net in Base 3 — Upper bound on s

There is no (75, 128, 1105)-net in base 3, because

1 times m-reduction [i] would yield (75, 127, 1105)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 016267 378504 014261 640293 630128 139598 020725 046986 807412 772361 > 3¹²⁷ [i]