MinT - Best Known (109−68, 109, s)-Nets in Base 3

Best Known (109−68, 109, s)-Nets in Base 3

(109−68, 109, 42)-Net over F₃ — Constructive and digital

Digital (41, 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]

(109−68, 109, 56)-Net over F₃ — Digital

Digital (41, 109, 56)-net over F₃, using

t-expansion [i] based on 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]

(109−68, 109, 142)-Net in Base 3 — Upper bound on s

There is no (41, 109, 143)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁰⁹, 143, S₃, 68), but
- the linear programming bound shows that MÂ â‰¥ 1 396002 643512 427111 634648 557176 258845 666621 442835 591463 756628 549772 300641 / 131 634445 751877 506312 > 3¹⁰⁹ [i]