MinT - Best Known (211−123, 211, s)-Nets in Base 3

Best Known (211−123, 211, s)-Nets in Base 3

(211−123, 211, 63)-Net over F₃ — Constructive and digital

Digital (88, 211, 63)-net over F₃, using

net from sequence [i] based on digital (88, 62)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 62)-sequence over F₉, using
  - s-reduction 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]

(211−123, 211, 84)-Net over F₃ — Digital

Digital (88, 211, 84)-net over F₃, using

t-expansion [i] based on digital (71, 211, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(211−123, 211, 459)-Net in Base 3 — Upper bound on s

There is no (88, 211, 460)-net in base 3, because

1 times m-reduction [i] would yield (88, 210, 460)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16506 184235 983148 800222 657447 398298 624581 250814 471081 392018 701902 287830 837605 214901 314376 966519 218297 > 3²¹⁰ [i]