MinT - Best Known (223−127, 223, s)-Nets in Base 3

Best Known (223−127, 223, s)-Nets in Base 3

(223−127, 223, 64)-Net over F₃ — Constructive and digital

Digital (96, 223, 64)-net over F₃, using

t-expansion [i] based on digital (89, 223, 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]

(223−127, 223, 96)-Net over F₃ — Digital

Digital (96, 223, 96)-net over F₃, using

t-expansion [i] based on digital (89, 223, 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]

(223−127, 223, 523)-Net in Base 3 — Upper bound on s

There is no (96, 223, 524)-net in base 3, because

1 times m-reduction [i] would yield (96, 222, 524)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8818 507716 395964 326429 357680 744735 702887 813754 418132 400438 543796 953725 814592 902993 821748 675985 219637 463825 > 3²²² [i]