MinT - Best Known (108−68, 108, s)-Nets in Base 3

Best Known (108−68, 108, s)-Nets in Base 3

(108−68, 108, 42)-Net over F₃ — Constructive and digital

Digital (40, 108, 42)-net over F₃, using

t-expansion [i] based on digital (39, 108, 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]

(108−68, 108, 56)-Net over F₃ — Digital

Digital (40, 108, 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]

(108−68, 108, 137)-Net in Base 3 — Upper bound on s

There is no (40, 108, 138)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁰⁸, 138, S₃, 68), but
- the linear programming bound shows that MÂ â‰¥ 323 722056 937510 506810 705120 169488 750600 798324 126092 309772 715766 758497 / 75548 018704 324936 > 3¹⁰⁸ [i]