MinT - Best Known (137−53, 137, s)-Nets in Base 3

Best Known (137−53, 137, s)-Nets in Base 3

(137−53, 137, 128)-Net over F₃ — Constructive and digital

Digital (84, 137, 128)-net over F₃, using

5 times m-reduction [i] based on digital (84, 142, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 71, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(137−53, 137, 149)-Net over F₃ — Digital

Digital (84, 137, 149)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−53, 137, 1626)-Net in Base 3 — Upper bound on s

There is no (84, 137, 1627)-net in base 3, because

1 times m-reduction [i] would yield (84, 136, 1627)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 77933 518518 318197 566295 165647 280627 030311 384545 003679 430747 793765 > 3¹³⁶ [i]