MinT - Best Known (111, 111+121, s)-Nets in Base 3

Best Known (111, 111+121, s)-Nets in Base 3

(111, 111+121, 74)-Net over F₃ — Constructive and digital

Digital (111, 232, 74)-net over F₃, using

t-expansion [i] based on digital (107, 232, 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]

(111, 111+121, 104)-Net over F₃ — Digital

Digital (111, 232, 104)-net over F₃, using

t-expansion [i] based on digital (102, 232, 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]

(111, 111+121, 738)-Net in Base 3 — Upper bound on s

There is no (111, 232, 739)-net in base 3, because

1 times m-reduction [i] would yield (111, 231, 739)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 165 957653 752799 091404 968435 582975 567785 917718 613684 734618 949934 946723 975286 088326 158081 018233 124981 502873 314537 > 3²³¹ [i]