MinT - Best Known (40, 40+209, s)-Nets in Base 3

Best Known (40, 40+209, s)-Nets in Base 3

(40, 40+209, 42)-Net over F₃ — Constructive and digital

Digital (40, 249, 42)-net over F₃, using

t-expansion [i] based on digital (39, 249, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(40, 40+209, 56)-Net over F₃ — Digital

Digital (40, 249, 56)-net over F₃, using

net from sequence [i] based on digital (40, 55)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
  - a curve from the manYPoints database [i]

(40, 40+209, 100)-Net in Base 3 — Upper bound on s

There is no (40, 249, 101)-net in base 3, because

52 times m-reduction [i] would yield (40, 197, 101)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁹⁷, 101, S₃, 2, 157), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 062455 955503 499077 355125 288143 118507 316933 810613 577983 898299 846956 369963 605208 731977 538796 176004 / 79 > 3¹⁹⁷ [i]