MinT - Best Known (39, 39+73, s)-Nets in Base 3

Best Known (39, 39+73, s)-Nets in Base 3

(39, 39+73, 42)-Net over F₃ — Constructive and digital

Digital (39, 112, 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]

(39, 39+73, 52)-Net over F₃ — Digital

Digital (39, 112, 52)-net over F₃, using

t-expansion [i] based on digital (37, 112, 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]

(39, 39+73, 126)-Net in Base 3 — Upper bound on s

There is no (39, 112, 127)-net in base 3, because

1 times m-reduction [i] would yield (39, 111, 127)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹¹, 127, S₃, 72), but
  - the linear programming bound shows that MÂ â‰¥ 1 993002 744270 942199 588408 245235 913938 365234 057812 071961 845221 / 20 511029 > 3¹¹¹ [i]