MinT - Best Known (39, 39+69, s)-Nets in Base 3

Best Known (39, 39+69, s)-Nets in Base 3

(39, 39+69, 42)-Net over F₃ — Constructive and digital

Digital (39, 108, 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+69, 52)-Net over F₃ — Digital

Digital (39, 108, 52)-net over F₃, using

t-expansion [i] based on digital (37, 108, 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+69, 130)-Net in Base 3 — Upper bound on s

There is no (39, 108, 131)-net in base 3, because

1 times m-reduction [i] would yield (39, 107, 131)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁷, 131, S₃, 68), but
  - the linear programming bound shows that MÂ â‰¥ 10193 910511 382960 109087 704226 878170 396646 633604 029190 685770 555639 / 8 455854 494087 > 3¹⁰⁷ [i]