MinT - Best Known (37, 37+189, s)-Nets in Base 3

Best Known (37, 37+189, s)-Nets in Base 3

(37, 37+189, 38)-Net over F₃ — Constructive and digital

Digital (37, 226, 38)-net over F₃, using

t-expansion [i] based on digital (32, 226, 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]

(37, 37+189, 52)-Net over F₃ — Digital

Digital (37, 226, 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]

(37, 37+189, 93)-Net in Base 3 — Upper bound on s

There is no (37, 226, 94)-net in base 3, because

43 times m-reduction [i] would yield (37, 183, 94)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁸³, 94, S₃, 2, 146), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 18511 095575 145533 338447 403836 109581 505353 226740 782032 803932 380476 024584 374357 028238 763043 / 7 > 3¹⁸³ [i]