MinT - Best Known (89, 89+51, s)-Nets in Base 3

Best Known (89, 89+51, s)-Nets in Base 3

(89, 89+51, 148)-Net over F₃ — Constructive and digital

Digital (89, 140, 148)-net over F₃, using

4 times m-reduction [i] based on digital (89, 144, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 72, 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]

(89, 89+51, 180)-Net over F₃ — Digital

Digital (89, 140, 180)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+51, 2263)-Net in Base 3 — Upper bound on s

There is no (89, 140, 2264)-net in base 3, because

1 times m-reduction [i] would yield (89, 139, 2264)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 105522 263220 396999 644826 667539 076400 161095 480518 478168 423967 213873 > 3¹³⁹ [i]