MinT - Best Known (56, 56+35, s)-Nets in Base 3

Best Known (56, 56+35, s)-Nets in Base 3

(56, 56+35, 80)-Net over F₃ — Constructive and digital

Digital (56, 91, 80)-net over F₃, using

5 times m-reduction [i] based on digital (56, 96, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 48, 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]

(56, 56+35, 110)-Net over F₃ — Digital

Digital (56, 91, 110)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+35, 1188)-Net in Base 3 — Upper bound on s

There is no (56, 91, 1189)-net in base 3, because

1 times m-reduction [i] would yield (56, 90, 1189)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 824885 371627 953678 554001 561867 503507 630347 > 3⁹⁰ [i]