MinT - Best Known (109−71, 109, s)-Nets in Base 3

Best Known (109−71, 109, s)-Nets in Base 3

(109−71, 109, 38)-Net over F₃ — Constructive and digital

Digital (38, 109, 38)-net over F₃, using

t-expansion [i] based on digital (32, 109, 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]

(109−71, 109, 52)-Net over F₃ — Digital

Digital (38, 109, 52)-net over F₃, using

t-expansion [i] based on digital (37, 109, 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]

(109−71, 109, 123)-Net in Base 3 — Upper bound on s

There is no (38, 109, 124)-net in base 3, because

1 times m-reduction [i] would yield (38, 108, 124)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁸, 124, S₃, 70), but
  - the linear programming bound shows that MÂ â‰¥ 37 454952 569242 421982 996691 395370 215650 157240 469539 761140 263717 / 10878 767680 > 3¹⁰⁸ [i]