MinT - Best Known (77, 77+57, s)-Nets in Base 3

Best Known (77, 77+57, s)-Nets in Base 3

(77, 77+57, 80)-Net over F₃ — Constructive and digital

Digital (77, 134, 80)-net over F₃, using

4 times m-reduction [i] based on digital (77, 138, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 69, 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]

(77, 77+57, 112)-Net over F₃ — Digital

Digital (77, 134, 112)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+57, 1015)-Net in Base 3 — Upper bound on s

There is no (77, 134, 1016)-net in base 3, because

1 times m-reduction [i] would yield (77, 133, 1016)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2877 446283 453754 533015 680524 271121 676893 508901 243941 726145 058241 > 3¹³³ [i]