MinT - Best Known (180−135, 180, s)-Nets in Base 3

Best Known (180−135, 180, s)-Nets in Base 3

(180−135, 180, 48)-Net over F₃ — Constructive and digital

Digital (45, 180, 48)-net over F₃, using

net from sequence [i] based on digital (45, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(180−135, 180, 56)-Net over F₃ — Digital

Digital (45, 180, 56)-net over F₃, using

t-expansion [i] based on digital (40, 180, 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]

(180−135, 180, 143)-Net in Base 3 — Upper bound on s

There is no (45, 180, 144)-net in base 3, because

53 times m-reduction [i] would yield (45, 127, 144)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹²⁷, 144, S₃, 82), but
  - the linear programming bound shows that MÂ â‰¥ 13577 961473 879661 811848 237829 138963 029891 906866 238711 650620 748976 085767 / 3257 508055 > 3¹²⁷ [i]