MinT - Best Known (86, 182, s)-Nets in Base 3

Best Known (86, 182, s)-Nets in Base 3

(86, 182, 61)-Net over F₃ — Constructive and digital

Digital (86, 182, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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]

(86, 182, 84)-Net over F₃ — Digital

Digital (86, 182, 84)-net over F₃, using

t-expansion [i] based on digital (71, 182, 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]

(86, 182, 557)-Net in Base 3 — Upper bound on s

There is no (86, 182, 558)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 700 695010 537308 269995 703411 400867 949457 957724 940512 258505 630354 605407 983019 536117 753569 > 3¹⁸² [i]