MinT - Best Known (232−123, 232, s)-Nets in Base 3

Best Known (232−123, 232, s)-Nets in Base 3

(232−123, 232, 74)-Net over F₃ — Constructive and digital

Digital (109, 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]

(232−123, 232, 104)-Net over F₃ — Digital

Digital (109, 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]

(232−123, 232, 696)-Net in Base 3 — Upper bound on s

There is no (109, 232, 697)-net in base 3, because

1 times m-reduction [i] would yield (109, 231, 697)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 170 585188 681946 291944 923658 391047 119926 422523 557927 610689 555083 971605 683281 893298 727395 438358 832570 236192 953475 > 3²³¹ [i]