MinT - Best Known (247−131, 247, s)-Nets in Base 3

Best Known (247−131, 247, s)-Nets in Base 3

(247−131, 247, 74)-Net over F₃ — Constructive and digital

Digital (116, 247, 74)-net over F₃, using

t-expansion [i] based on digital (107, 247, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [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]

(247−131, 247, 120)-Net over F₃ — Digital

Digital (116, 247, 120)-net over F₃, using

t-expansion [i] based on digital (113, 247, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(247−131, 247, 738)-Net in Base 3 — Upper bound on s

There is no (116, 247, 739)-net in base 3, because

1 times m-reduction [i] would yield (116, 246, 739)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2522 468375 934139 539877 577185 803745 303747 886158 976563 584636 329034 523925 784679 318323 734522 890277 190047 340166 227993 184071 > 3²⁴⁶ [i]