MinT - Best Known (198−108, 198, s)-Nets in Base 3

Best Known (198−108, 198, s)-Nets in Base 3

(198−108, 198, 64)-Net over F₃ — Constructive and digital

Digital (90, 198, 64)-net over F₃, using

t-expansion [i] based on digital (89, 198, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(198−108, 198, 96)-Net over F₃ — Digital

Digital (90, 198, 96)-net over F₃, using

t-expansion [i] based on digital (89, 198, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(198−108, 198, 537)-Net in Base 3 — Upper bound on s

There is no (90, 198, 538)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 31716 674208 144067 279994 552252 818373 160094 927544 746163 917417 064631 485932 743941 548027 329339 227949 > 3¹⁹⁸ [i]