MinT - Best Known (249−144, 249, s)-Nets in Base 3

Best Known (249−144, 249, s)-Nets in Base 3

(249−144, 249, 72)-Net over F₃ — Constructive and digital

Digital (105, 249, 72)-net over F₃, using

net from sequence [i] based on digital (105, 71)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 71)-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]

(249−144, 249, 104)-Net over F₃ — Digital

Digital (105, 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]

(249−144, 249, 549)-Net in Base 3 — Upper bound on s

There is no (105, 249, 550)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 71039 080961 590701 464533 996295 188595 979851 255878 360379 685061 683660 627940 744253 210828 730064 028587 501339 872055 293302 338641 > 3²⁴⁹ [i]