MinT - Best Known (126−69, 126, s)-Nets in Base 3

Best Known (126−69, 126, s)-Nets in Base 3

(126−69, 126, 48)-Net over F₃ — Constructive and digital

Digital (57, 126, 48)-net over F₃, using

t-expansion [i] based on digital (45, 126, 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]

(126−69, 126, 64)-Net over F₃ — Digital

Digital (57, 126, 64)-net over F₃, using

t-expansion [i] based on digital (49, 126, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(126−69, 126, 351)-Net in Base 3 — Upper bound on s

There is no (57, 126, 352)-net in base 3, because

1 times m-reduction [i] would yield (57, 125, 352)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 441606 703895 197218 544912 205976 089273 258244 390760 635534 678081 > 3¹²⁵ [i]