MinT - Best Known (42, 42+69, s)-Nets in Base 3

Best Known (42, 42+69, s)-Nets in Base 3

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

Digital (42, 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]

(42, 42+69, 56)-Net over F₃ — Digital

Digital (42, 111, 56)-net over F₃, using

t-expansion [i] based on 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]

(42, 42+69, 147)-Net in Base 3 — Upper bound on s

There is no (42, 111, 148)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹¹¹, 148, S₃, 69), but
- the linear programming bound shows that MÂ â‰¥ 34269 689678 954827 771722 528323 125817 537099 985988 010230 663767 339699 553771 239936 / 338753 436767 266709 313125 > 3¹¹¹ [i]