MinT - Best Known (86, 86+73, s)-Nets in Base 3

Best Known (86, 86+73, s)-Nets in Base 3

(86, 86+73, 69)-Net over F₃ — Constructive and digital

Digital (86, 159, 69)-net over F₃, using

3 times m-reduction [i] based on digital (86, 162, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 59, 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, 103, 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]

(86, 86+73, 104)-Net over F₃ — Digital

Digital (86, 159, 104)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+73, 851)-Net in Base 3 — Upper bound on s

There is no (86, 159, 852)-net in base 3, because

1 times m-reduction [i] would yield (86, 158, 852)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2456 837565 135313 521957 037697 410390 056494 893490 591239 516990 723897 513619 644225 > 3¹⁵⁸ [i]