MinT - Best Known (42, 42+78, s)-Nets in Base 3

Best Known (42, 42+78, s)-Nets in Base 3

(42, 42+78, 42)-Net over F₃ — Constructive and digital

Digital (42, 120, 42)-net over F₃, using

t-expansion [i] based on digital (39, 120, 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+78, 56)-Net over F₃ — Digital

Digital (42, 120, 56)-net over F₃, using

t-expansion [i] based on digital (40, 120, 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+78, 135)-Net in Base 3 — Upper bound on s

There is no (42, 120, 136)-net in base 3, because

1 times m-reduction [i] would yield (42, 119, 136)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁹, 136, S₃, 77), but
  - the linear programming bound shows that MÂ â‰¥ 2 432185 386357 114583 869869 341707 927063 024092 591158 990986 554502 987729 / 3299 392408 > 3¹¹⁹ [i]