MinT - Best Known (56, 56+37, s)-Nets in Base 3

Best Known (56, 56+37, s)-Nets in Base 3

(56, 56+37, 80)-Net over F₃ — Constructive and digital

Digital (56, 93, 80)-net over F₃, using

3 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+37, 101)-Net over F₃ — Digital

Digital (56, 93, 101)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+37, 1019)-Net in Base 3 — Upper bound on s

There is no (56, 93, 1020)-net in base 3, because

1 times m-reduction [i] would yield (56, 92, 1020)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 79 261442 424289 697179 189737 852767 084071 534345 > 3⁹² [i]