MinT - Best Known (210−119, 210, s)-Nets in Base 3

Best Known (210−119, 210, s)-Nets in Base 3

(210−119, 210, 64)-Net over F₃ — Constructive and digital

Digital (91, 210, 64)-net over F₃, using

t-expansion [i] based on digital (89, 210, 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]

(210−119, 210, 96)-Net over F₃ — Digital

Digital (91, 210, 96)-net over F₃, using

t-expansion [i] based on digital (89, 210, 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]

(210−119, 210, 502)-Net in Base 3 — Upper bound on s

There is no (91, 210, 503)-net in base 3, because

1 times m-reduction [i] would yield (91, 209, 503)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5248 378831 103000 880070 761771 681990 406340 185410 320679 336871 701697 841630 697966 663046 376501 726651 806443 > 3²⁰⁹ [i]