MinT - Best Known (56−19, 56, s)-Nets in Base 3

Best Known (56−19, 56, s)-Nets in Base 3

(56−19, 56, 80)-Net over F₃ — Constructive and digital

Digital (37, 56, 80)-net over F₃, using

2 times m-reduction [i] based on digital (37, 58, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 29, 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]

(56−19, 56, 113)-Net over F₃ — Digital

Digital (37, 56, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−19, 56, 1699)-Net in Base 3 — Upper bound on s

There is no (37, 56, 1700)-net in base 3, because

1 times m-reduction [i] would yield (37, 55, 1700)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 174 985175 002170 402957 892681 > 3⁵⁵ [i]