MinT - Best Known (89−39, 89, s)-Nets in Base 3

Best Known (89−39, 89, s)-Nets in Base 3

(89−39, 89, 64)-Net over F₃ — Constructive and digital

Digital (50, 89, 64)-net over F₃, using

1 times m-reduction [i] based on digital (50, 90, 64)-net over F₃, using
- trace code for nets [i] based on digital (5, 45, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]

(89−39, 89, 75)-Net over F₃ — Digital

Digital (50, 89, 75)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−39, 89, 624)-Net in Base 3 — Upper bound on s

There is no (50, 89, 625)-net in base 3, because

1 times m-reduction [i] would yield (50, 88, 625)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 986474 021512 942660 689588 114451 984080 052251 > 3⁸⁸ [i]