MinT - Best Known (127−57, 127, s)-Nets in Base 3

Best Known (127−57, 127, s)-Nets in Base 3

(127−57, 127, 68)-Net over F₃ — Constructive and digital

Digital (70, 127, 68)-net over F₃, using

1 times m-reduction [i] based on digital (70, 128, 68)-net over F₃, using
- trace code for nets [i] based on digital (6, 64, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]

(127−57, 127, 92)-Net over F₃ — Digital

Digital (70, 127, 92)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−57, 127, 765)-Net in Base 3 — Upper bound on s

There is no (70, 127, 766)-net in base 3, because

1 times m-reduction [i] would yield (70, 126, 766)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 336654 612487 606293 298343 101010 982943 186322 783237 120034 407801 > 3¹²⁶ [i]