MinT - Best Known (172−138, 172, s)-Nets in Base 3

Best Known (172−138, 172, s)-Nets in Base 3

(172−138, 172, 38)-Net over F₃ — Constructive and digital

Digital (34, 172, 38)-net over F₃, using

t-expansion [i] based on digital (32, 172, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(172−138, 172, 46)-Net over F₃ — Digital

Digital (34, 172, 46)-net over F₃, using

t-expansion [i] based on digital (33, 172, 46)-net over F₃, using
- net from sequence [i] based on digital (33, 45)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
    - a curve from the manYPoints database [i]

(172−138, 172, 86)-Net in Base 3 — Upper bound on s

There is no (34, 172, 87)-net in base 3, because

3 times m-reduction [i] would yield (34, 169, 87)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁹, 87, S₃, 2, 135), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 34831 892110 592771 988701 292967 816900 663202 927024 414813 316873 143916 304132 301341 123923 / 68 > 3¹⁶⁹ [i]