MinT - Best Known (198−65, 198, s)-Nets in Base 3

Best Known (198−65, 198, s)-Nets in Base 3

(198−65, 198, 162)-Net over F₃ — Constructive and digital

Digital (133, 198, 162)-net over F₃, using

4 times m-reduction [i] based on digital (133, 202, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 101, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(198−65, 198, 331)-Net over F₃ — Digital

Digital (133, 198, 331)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−65, 198, 5503)-Net in Base 3 — Upper bound on s

There is no (133, 198, 5504)-net in base 3, because

1 times m-reduction [i] would yield (133, 197, 5504)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9854 169511 517045 941810 580220 807852 574962 903044 743009 029245 857677 217584 147875 215963 401377 595393 > 3¹⁹⁷ [i]