MinT - Best Known (164−77, 164, s)-Nets in Base 3

Best Known (164−77, 164, s)-Nets in Base 3

(164−77, 164, 69)-Net over F₃ — Constructive and digital

Digital (87, 164, 69)-net over F₃, using

1 times m-reduction [i] based on digital (87, 165, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 60, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]
  2. digital (27, 105, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(164−77, 164, 101)-Net over F₃ — Digital

Digital (87, 164, 101)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−77, 164, 799)-Net in Base 3 — Upper bound on s

There is no (87, 164, 800)-net in base 3, because

1 times m-reduction [i] would yield (87, 163, 800)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 603318 355318 158294 825495 531475 849370 880780 526763 567740 186393 677135 539748 665025 > 3¹⁶³ [i]