MinT - Best Known (41, 41+72, s)-Nets in Base 3

Best Known (41, 41+72, s)-Nets in Base 3

(41, 41+72, 42)-Net over F₃ — Constructive and digital

Digital (41, 113, 42)-net over F₃, using

t-expansion [i] based on digital (39, 113, 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]

(41, 41+72, 56)-Net over F₃ — Digital

Digital (41, 113, 56)-net over F₃, using

t-expansion [i] based on digital (40, 113, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(41, 41+72, 135)-Net in Base 3 — Upper bound on s

There is no (41, 113, 136)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹¹³, 136, S₃, 72), but
- the linear programming bound shows that MÂ â‰¥ 1291 389105 848107 529601 644247 240146 488912 309793 168781 016295 163432 / 1517 040959 > 3¹¹³ [i]