MinT - Best Known (41, 41+70, s)-Nets in Base 3

Best Known (41, 41+70, s)-Nets in Base 3

(41, 41+70, 42)-Net over F₃ — Constructive and digital

Digital (41, 111, 42)-net over F₃, using

t-expansion [i] based on digital (39, 111, 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+70, 56)-Net over F₃ — Digital

Digital (41, 111, 56)-net over F₃, using

t-expansion [i] based on digital (40, 111, 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+70, 140)-Net in Base 3 — Upper bound on s

There is no (41, 111, 141)-net in base 3, because

1 times m-reduction [i] would yield (41, 110, 141)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁰, 141, S₃, 69), but
  - the linear programming bound shows that MÂ â‰¥ 2111 324064 305608 092070 313484 417163 913990 086868 398347 464858 249900 635788 / 60842 907930 162995 > 3¹¹⁰ [i]