MinT - Best Known (69, 69+65, s)-Nets in Base 3

Best Known (69, 69+65, s)-Nets in Base 3

(69, 69+65, 60)-Net over F₃ — Constructive and digital

Digital (69, 134, 60)-net over F₃, using

1 times m-reduction [i] based on digital (69, 135, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 48, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (21, 87, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]

(69, 69+65, 82)-Net over F₃ — Digital

Digital (69, 134, 82)-net over F₃, using

net from sequence [i] based on digital (69, 81)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
  - Drinfeld modules of rank 1 [i]

(69, 69+65, 584)-Net in Base 3 — Upper bound on s

There is no (69, 134, 585)-net in base 3, because

1 times m-reduction [i] would yield (69, 133, 585)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2991 460851 310676 756240 618370 811121 276914 640005 603914 903979 992705 > 3¹³³ [i]