MinT - Best Known (65−27, 65, s)-Nets in Base 3

Best Known (65−27, 65, s)-Nets in Base 3

(65−27, 65, 64)-Net over F₃ — Constructive and digital

Digital (38, 65, 64)-net over F₃, using

1 times m-reduction [i] based on digital (38, 66, 64)-net over F₃, using
- trace code for nets [i] based on digital (5, 33, 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]

(65−27, 65, 69)-Net over F₃ — Digital

Digital (38, 65, 69)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−27, 65, 620)-Net in Base 3 — Upper bound on s

There is no (38, 65, 621)-net in base 3, because

1 times m-reduction [i] would yield (38, 64, 621)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 474588 383485 053577 500015 039307 > 3⁶⁴ [i]