MinT - Best Known (87, 87+127, s)-Nets in Base 3

Best Known (87, 87+127, s)-Nets in Base 3

(87, 87+127, 62)-Net over F₃ — Constructive and digital

Digital (87, 214, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-sequence over F₉, using
  - s-reduction 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]

(87, 87+127, 84)-Net over F₃ — Digital

Digital (87, 214, 84)-net over F₃, using

t-expansion [i] based on digital (71, 214, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(87, 87+127, 439)-Net in Base 3 — Upper bound on s

There is no (87, 214, 440)-net in base 3, because

1 times m-reduction [i] would yield (87, 213, 440)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 470538 187169 158988 721700 395289 029603 944921 746543 063944 340220 664874 692594 781738 014823 362987 995093 360289 > 3²¹³ [i]