MinT - Best Known (199−121, 199, s)-Nets in Base 3

Best Known (199−121, 199, s)-Nets in Base 3

(199−121, 199, 53)-Net over F₃ — Constructive and digital

Digital (78, 199, 53)-net over F₃, using

net from sequence [i] based on digital (78, 52)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 52)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(199−121, 199, 84)-Net over F₃ — Digital

Digital (78, 199, 84)-net over F₃, using

t-expansion [i] based on digital (71, 199, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(199−121, 199, 378)-Net in Base 3 — Upper bound on s

There is no (78, 199, 379)-net in base 3, because

1 times m-reduction [i] would yield (78, 198, 379)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29624 642986 495946 654089 052217 936963 419850 291542 235384 107729 212199 047778 407932 623177 073143 168361 > 3¹⁹⁸ [i]