MinT - Best Known (117−76, 117, s)-Nets in Base 3

Best Known (117−76, 117, s)-Nets in Base 3

(117−76, 117, 42)-Net over F₃ — Constructive and digital

Digital (41, 117, 42)-net over F₃, using

t-expansion [i] based on digital (39, 117, 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]

(117−76, 117, 56)-Net over F₃ — Digital

Digital (41, 117, 56)-net over F₃, using

t-expansion [i] based on digital (40, 117, 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]

(117−76, 117, 132)-Net in Base 3 — Upper bound on s

There is no (41, 117, 133)-net in base 3, because

1 times m-reduction [i] would yield (41, 116, 133)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁶, 133, S₃, 75), but
  - the linear programming bound shows that MÂ â‰¥ 499 991160 796691 767829 311089 863604 958229 687121 227994 002589 482235 / 20 811757 > 3¹¹⁶ [i]