MinT - Best Known (137−57, 137, s)-Nets in Base 3

Best Known (137−57, 137, s)-Nets in Base 3

(137−57, 137, 80)-Net over F₃ — Constructive and digital

Digital (80, 137, 80)-net over F₃, using

7 times m-reduction [i] based on digital (80, 144, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 72, 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]

(137−57, 137, 121)-Net over F₃ — Digital

Digital (80, 137, 121)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−57, 137, 1146)-Net in Base 3 — Upper bound on s

There is no (80, 137, 1147)-net in base 3, because

1 times m-reduction [i] would yield (80, 136, 1147)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 79037 293564 840009 919891 026579 210255 872283 338318 293443 480993 213353 > 3¹³⁶ [i]