MinT - Best Known (42, 42+74, s)-Nets in Base 3

Best Known (42, 42+74, s)-Nets in Base 3

(42, 42+74, 42)-Net over F₃ — Constructive and digital

Digital (42, 116, 42)-net over F₃, using

t-expansion [i] based on digital (39, 116, 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]

(42, 42+74, 56)-Net over F₃ — Digital

Digital (42, 116, 56)-net over F₃, using

t-expansion [i] based on digital (40, 116, 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]

(42, 42+74, 138)-Net in Base 3 — Upper bound on s

There is no (42, 116, 139)-net in base 3, because

1 times m-reduction [i] would yield (42, 115, 139)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁵, 139, S₃, 73), but
  - the linear programming bound shows that MÂ â‰¥ 53 726456 298605 394882 640035 124713 478057 255728 840609 934833 221759 641680 / 6 655955 254091 > 3¹¹⁵ [i]