MinT - Best Known (199−69, 199, s)-Nets in Base 3

Best Known (199−69, 199, s)-Nets in Base 3

(199−69, 199, 156)-Net over F₃ — Constructive and digital

Digital (130, 199, 156)-net over F₃, using

17 times m-reduction [i] based on digital (130, 216, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 108, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(199−69, 199, 281)-Net over F₃ — Digital

Digital (130, 199, 281)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−69, 199, 4030)-Net in Base 3 — Upper bound on s

There is no (130, 199, 4031)-net in base 3, because

1 times m-reduction [i] would yield (130, 198, 4031)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29558 367778 834847 827632 547871 846026 542987 179212 810017 719605 282412 467832 017926 827974 156826 992445 > 3¹⁹⁸ [i]