MinT - Best Known (246−143, 246, s)-Nets in Base 3

Best Known (246−143, 246, s)-Nets in Base 3

(246−143, 246, 70)-Net over F₃ — Constructive and digital

Digital (103, 246, 70)-net over F₃, using

net from sequence [i] based on digital (103, 69)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 69)-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]

(246−143, 246, 104)-Net over F₃ — Digital

Digital (103, 246, 104)-net over F₃, using

t-expansion [i] based on digital (102, 246, 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]

(246−143, 246, 536)-Net in Base 3 — Upper bound on s

There is no (103, 246, 537)-net in base 3, because

1 times m-reduction [i] would yield (103, 245, 537)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 815 308679 130104 054463 366457 191111 343251 370344 865909 313968 022526 515320 554428 781936 593358 109764 794760 370165 166819 594475 > 3²⁴⁵ [i]