MinT - Best Known (42, 42+75, s)-Nets in Base 3

Best Known (42, 42+75, s)-Nets in Base 3

(42, 42+75, 42)-Net over F₃ — Constructive and digital

Digital (42, 117, 42)-net over F₃, using

t-expansion [i] based on digital (39, 117, 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+75, 56)-Net over F₃ — Digital

Digital (42, 117, 56)-net over F₃, using

t-expansion [i] based on digital (40, 117, 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+75, 136)-Net in Base 3 — Upper bound on s

There is no (42, 117, 137)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹¹⁷, 137, S₃, 75), but
- the linear programming bound shows that MÂ â‰¥ 2 512290 520419 287793 177886 934344 555139 842858 289208 378001 236620 873421 / 34498 790200 > 3¹¹⁷ [i]