MinT - Best Known (199−75, 199, s)-Nets in Base 3

Best Known (199−75, 199, s)-Nets in Base 3

(199−75, 199, 156)-Net over F₃ — Constructive and digital

Digital (124, 199, 156)-net over F₃, using

5 times m-reduction [i] based on digital (124, 204, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 102, 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]

(199−75, 199, 221)-Net over F₃ — Digital

Digital (124, 199, 221)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−75, 199, 2582)-Net in Base 3 — Upper bound on s

There is no (124, 199, 2583)-net in base 3, because

1 times m-reduction [i] would yield (124, 198, 2583)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29659 292238 325533 590409 600714 874708 976941 491153 777714 130691 030597 750900 435212 159147 932312 289703 > 3¹⁹⁸ [i]