MinT - Best Known (109−69, 109, s)-Nets in Base 3

Best Known (109−69, 109, s)-Nets in Base 3

(109−69, 109, 42)-Net over F₃ — Constructive and digital

Digital (40, 109, 42)-net over F₃, using

t-expansion [i] based on digital (39, 109, 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]

(109−69, 109, 56)-Net over F₃ — Digital

Digital (40, 109, 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]

(109−69, 109, 135)-Net in Base 3 — Upper bound on s

There is no (40, 109, 136)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁰⁹, 136, S₃, 69), but
- the linear programming bound shows that MÂ â‰¥ 1404 162922 486264 867350 267781 542823 568518 687474 356865 362339 807006 187977 / 102274 846983 013705 > 3¹⁰⁹ [i]