MinT - Best Known (172−125, 172, s)-Nets in Base 3

Best Known (172−125, 172, s)-Nets in Base 3

(172−125, 172, 48)-Net over F₃ — Constructive and digital

Digital (47, 172, 48)-net over F₃, using

t-expansion [i] based on digital (45, 172, 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]

(172−125, 172, 56)-Net over F₃ — Digital

Digital (47, 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]

(172−125, 172, 148)-Net in Base 3 — Upper bound on s

There is no (47, 172, 149)-net in base 3, because

36 times m-reduction [i] would yield (47, 136, 149)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹³⁶, 149, S₃, 89), but
  - the linear programming bound shows that MÂ â‰¥ 6 385896 362423 839893 456186 755923 270794 744764 572661 578917 513381 685836 804369 / 75 686875 > 3¹³⁶ [i]