MinT - Best Known (126−53, 126, s)-Nets in Base 3

Best Known (126−53, 126, s)-Nets in Base 3

(126−53, 126, 80)-Net over F₃ — Constructive and digital

Digital (73, 126, 80)-net over F₃, using

4 times m-reduction [i] based on digital (73, 130, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 65, 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]

(126−53, 126, 110)-Net over F₃ — Digital

Digital (73, 126, 110)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−53, 126, 1012)-Net in Base 3 — Upper bound on s

There is no (73, 126, 1013)-net in base 3, because

1 times m-reduction [i] would yield (73, 125, 1013)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 441489 870608 419294 812334 238007 356507 443195 533785 834593 247121 > 3¹²⁵ [i]