MinT - Best Known (39−19, 39, s)-Nets in Base 3

Best Known (39−19, 39, s)-Nets in Base 3

(39−19, 39, 29)-Net over F₃ — Constructive and digital

Digital (20, 39, 29)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 13)-net over F₃, using
  - a shift-net [i]
2. digital (7, 26, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]

(39−19, 39, 32)-Net over F₃ — Digital

Digital (20, 39, 32)-net over F₃, using

t-expansion [i] based on digital (19, 39, 32)-net over F₃, using
- net from sequence [i] based on digital (19, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 19 and N(F) ≥ 32, using
    - a curve from the manYPoints database [i]

(39−19, 39, 206)-Net in Base 3 — Upper bound on s

There is no (20, 39, 207)-net in base 3, because

1 times m-reduction [i] would yield (20, 38, 207)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 406127 512775 605135 > 3³⁸ [i]