MinT - Best Known (198−87, 198, s)-Nets in Base 3

Best Known (198−87, 198, s)-Nets in Base 3

(198−87, 198, 80)-Net over F₃ — Constructive and digital

Digital (111, 198, 80)-net over F₃, using

8 times m-reduction [i] based on digital (111, 206, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 103, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(198−87, 198, 142)-Net over F₃ — Digital

Digital (111, 198, 142)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−87, 198, 1253)-Net in Base 3 — Upper bound on s

There is no (111, 198, 1254)-net in base 3, because

1 times m-reduction [i] would yield (111, 197, 1254)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10095 556318 172344 112470 694566 517091 209373 220496 660063 224277 083918 734274 370532 496848 165427 413585 > 3¹⁹⁷ [i]