MinT - Best Known (218−127, 218, s)-Nets in Base 3

Best Known (218−127, 218, s)-Nets in Base 3

(218−127, 218, 64)-Net over F₃ — Constructive and digital

Digital (91, 218, 64)-net over F₃, using

t-expansion [i] based on digital (89, 218, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(218−127, 218, 96)-Net over F₃ — Digital

Digital (91, 218, 96)-net over F₃, using

t-expansion [i] based on digital (89, 218, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(218−127, 218, 474)-Net in Base 3 — Upper bound on s

There is no (91, 218, 475)-net in base 3, because

1 times m-reduction [i] would yield (91, 217, 475)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 310230 387221 150716 456703 617543 879217 352608 922119 586132 618651 937023 180568 376249 643833 988462 288504 673267 > 3²¹⁷ [i]