MinT - Best Known (111, 111+128, s)-Nets in Base 3

Best Known (111, 111+128, s)-Nets in Base 3

(111, 111+128, 74)-Net over F₃ — Constructive and digital

Digital (111, 239, 74)-net over F₃, using

t-expansion [i] based on digital (107, 239, 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+128, 104)-Net over F₃ — Digital

Digital (111, 239, 104)-net over F₃, using

t-expansion [i] based on digital (102, 239, 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+128, 685)-Net in Base 3 — Upper bound on s

There is no (111, 239, 686)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 169790 380351 257285 332429 823291 830457 870477 309315 148361 852482 799098 542334 562369 444869 520150 727684 057932 936002 127489 > 3²³⁹ [i]