MinT - Best Known (221−127, 221, s)-Nets in Base 3

Best Known (221−127, 221, s)-Nets in Base 3

(221−127, 221, 64)-Net over F₃ — Constructive and digital

Digital (94, 221, 64)-net over F₃, using

t-expansion [i] based on digital (89, 221, 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]

(221−127, 221, 96)-Net over F₃ — Digital

Digital (94, 221, 96)-net over F₃, using

t-expansion [i] based on digital (89, 221, 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]

(221−127, 221, 503)-Net in Base 3 — Upper bound on s

There is no (94, 221, 504)-net in base 3, because

1 times m-reduction [i] would yield (94, 220, 504)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 971 513699 799387 141126 227817 745829 251460 407205 468984 307186 335759 621309 406912 939223 600022 425335 692861 227425 > 3²²⁰ [i]