MinT - Best Known (172−131, 172, s)-Nets in Base 3

Best Known (172−131, 172, s)-Nets in Base 3

(172−131, 172, 42)-Net over F₃ — Constructive and digital

Digital (41, 172, 42)-net over F₃, using

t-expansion [i] based on digital (39, 172, 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]

(172−131, 172, 56)-Net over F₃ — Digital

Digital (41, 172, 56)-net over F₃, using

t-expansion [i] based on digital (40, 172, 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]

(172−131, 172, 131)-Net in Base 3 — Upper bound on s

There is no (41, 172, 132)-net in base 3, because

53 times m-reduction [i] would yield (41, 119, 132)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁹, 132, S₃, 78), but
  - the linear programming bound shows that MÂ â‰¥ 23379 936017 655610 429125 890884 181024 514039 893587 482992 919339 273663 / 33 418343 > 3¹¹⁹ [i]