MinT - Best Known (138−51, 138, s)-Nets in Base 3

Best Known (138−51, 138, s)-Nets in Base 3

(138−51, 138, 148)-Net over F₃ — Constructive and digital

Digital (87, 138, 148)-net over F₃, using

2 times m-reduction [i] based on digital (87, 140, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 70, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(138−51, 138, 171)-Net over F₃ — Digital

Digital (87, 138, 171)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−51, 138, 2070)-Net in Base 3 — Upper bound on s

There is no (87, 138, 2071)-net in base 3, because

1 times m-reduction [i] would yield (87, 137, 2071)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 232637 220263 976602 275566 516145 749371 715166 855199 439499 557018 615807 > 3¹³⁷ [i]