MinT - Best Known (69, 69+31, s)-Nets in Base 3

Best Known (69, 69+31, s)-Nets in Base 3

(69, 69+31, 148)-Net over F₃ — Constructive and digital

Digital (69, 100, 148)-net over F₃, using

4 times m-reduction [i] based on digital (69, 104, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 52, 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]

(69, 69+31, 227)-Net over F₃ — Digital

Digital (69, 100, 227)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+31, 4511)-Net in Base 3 — Upper bound on s

There is no (69, 100, 4512)-net in base 3, because

1 times m-reduction [i] would yield (69, 99, 4512)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 171933 394068 114300 259632 443804 203458 293436 955521 > 3⁹⁹ [i]