MinT - Best Known (40, 111, s)-Nets in Base 3

Best Known (40, 111, s)-Nets in Base 3

(40, 111, 42)-Net over F₃ — Constructive and digital

Digital (40, 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]

(40, 111, 56)-Net over F₃ — Digital

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]

(40, 111, 133)-Net in Base 3 — Upper bound on s

There is no (40, 111, 134)-net in base 3, because

1 times m-reduction [i] would yield (40, 110, 134)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁰, 134, S₃, 70), but
  - the linear programming bound shows that MÂ â‰¥ 56 614736 193362 990874 674864 143855 321591 421753 879494 724955 603321 584007 / 1414 070541 496450 > 3¹¹⁰ [i]