MinT - Best Known (210−172, 210, s)-Nets in Base 3

Best Known (210−172, 210, s)-Nets in Base 3

(210−172, 210, 38)-Net over F₃ — Constructive and digital

Digital (38, 210, 38)-net over F₃, using

t-expansion [i] based on digital (32, 210, 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]

(210−172, 210, 52)-Net over F₃ — Digital

Digital (38, 210, 52)-net over F₃, using

t-expansion [i] based on digital (37, 210, 52)-net over F₃, using
- net from sequence [i] based on digital (37, 51)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
    - a curve from the manYPoints database [i]

(210−172, 210, 95)-Net in Base 3 — Upper bound on s

There is no (38, 210, 96)-net in base 3, because

23 times m-reduction [i] would yield (38, 187, 96)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁸⁷, 96, S₃, 2, 149), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 4 498196 224760 364601 242719 132174 628305 800834 098010 033971 355568 455673 974002 968757 862019 419449 / 25 > 3¹⁸⁷ [i]