MinT - Best Known (89, 242, s)-Nets in Base 3

Best Known (89, 242, s)-Nets in Base 3

(89, 242, 64)-Net over F₃ — Constructive and digital

Digital (89, 242, 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]

(89, 242, 96)-Net over F₃ — Digital

Digital (89, 242, 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]

(89, 242, 403)-Net in Base 3 — Upper bound on s

There is no (89, 242, 404)-net in base 3, because

1 times m-reduction [i] would yield (89, 241, 404)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 578316 995226 224044 825762 433749 757244 038692 176356 952397 160496 338492 210288 074953 767388 259107 757346 449233 602728 745089 > 3²⁴¹ [i]