MinT - Best Known (172−81, 172, s)-Nets in Base 3

Best Known (172−81, 172, s)-Nets in Base 3

(172−81, 172, 69)-Net over F₃ — Constructive and digital

Digital (91, 172, 69)-net over F₃, using

5 times m-reduction [i] based on digital (91, 177, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 64, 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, 113, 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]

(172−81, 172, 104)-Net over F₃ — Digital

Digital (91, 172, 104)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−81, 172, 825)-Net in Base 3 — Upper bound on s

There is no (91, 172, 826)-net in base 3, because

1 times m-reduction [i] would yield (91, 171, 826)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4031 696337 358485 581896 345377 361369 434107 584374 263498 508144 558934 968107 145024 215313 > 3¹⁷¹ [i]