MinT - Best Known (139−75, 139, s)-Nets in Base 3

Best Known (139−75, 139, s)-Nets in Base 3

(139−75, 139, 48)-Net over F₃ — Constructive and digital

Digital (64, 139, 48)-net over F₃, using

t-expansion [i] based on digital (45, 139, 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]

(139−75, 139, 64)-Net over F₃ — Digital

Digital (64, 139, 64)-net over F₃, using

t-expansion [i] based on digital (49, 139, 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]

(139−75, 139, 405)-Net in Base 3 — Upper bound on s

There is no (64, 139, 406)-net in base 3, because

1 times m-reduction [i] would yield (64, 138, 406)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 715027 611502 064954 469405 525366 667900 777710 044334 649772 176325 750085 > 3¹³⁸ [i]