MinT - Best Known (249−141, 249, s)-Nets in Base 3

Best Known (249−141, 249, s)-Nets in Base 3

(249−141, 249, 74)-Net over F₃ — Constructive and digital

Digital (108, 249, 74)-net over F₃, using

t-expansion [i] based on digital (107, 249, 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]

(249−141, 249, 104)-Net over F₃ — Digital

Digital (108, 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]

(249−141, 249, 592)-Net in Base 3 — Upper bound on s

There is no (108, 249, 593)-net in base 3, because

1 times m-reduction [i] would yield (108, 248, 593)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22083 063718 316838 998500 753570 564566 542248 625744 616622 003720 683454 439782 399996 950479 110332 856327 791496 952343 085644 590841 > 3²⁴⁸ [i]