MinT - Best Known (249−145, 249, s)-Nets in Base 3

Best Known (249−145, 249, s)-Nets in Base 3

(249−145, 249, 71)-Net over F₃ — Constructive and digital

Digital (104, 249, 71)-net over F₃, using

net from sequence [i] based on digital (104, 70)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 70)-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]

(249−145, 249, 104)-Net over F₃ — Digital

Digital (104, 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]

(249−145, 249, 539)-Net in Base 3 — Upper bound on s

There is no (104, 249, 540)-net in base 3, because

1 times m-reduction [i] would yield (104, 248, 540)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21827 908685 694190 404254 056502 605584 129183 841537 834530 633615 022866 271041 377115 138233 614589 848778 566779 303629 200041 027009 > 3²⁴⁸ [i]