MinT - Best Known (249−137, 249, s)-Nets in Base 3

Best Known (249−137, 249, s)-Nets in Base 3

(249−137, 249, 74)-Net over F₃ — Constructive and digital

Digital (112, 249, 74)-net over F₃, using

t-expansion [i] based on digital (107, 249, 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]

(249−137, 249, 104)-Net over F₃ — Digital

Digital (112, 249, 104)-net over F₃, using

t-expansion [i] based on digital (102, 249, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(249−137, 249, 653)-Net in Base 3 — Upper bound on s

There is no (112, 249, 654)-net in base 3, because

1 times m-reduction [i] would yield (112, 248, 654)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21512 553468 077050 007400 080121 411925 380933 614896 834772 575298 894745 643776 860507 372972 621729 851450 659617 357192 195181 553545 > 3²⁴⁸ [i]