MinT - Best Known (239−140, 239, s)-Nets in Base 3

Best Known (239−140, 239, s)-Nets in Base 3

(239−140, 239, 66)-Net over F₃ — Constructive and digital

Digital (99, 239, 66)-net over F₃, using

net from sequence [i] based on digital (99, 65)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 65)-sequence over F₉, using
  - s-reduction 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]

(239−140, 239, 96)-Net over F₃ — Digital

Digital (99, 239, 96)-net over F₃, using

t-expansion [i] based on digital (89, 239, 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]

(239−140, 239, 506)-Net in Base 3 — Upper bound on s

There is no (99, 239, 507)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 189921 632637 753056 232420 113372 089214 277525 692318 225878 711398 853483 201265 208941 306389 019452 235965 831799 243722 665293 > 3²³⁹ [i]