MinT - Best Known (69, 69+72, s)-Nets in Base 3

Best Known (69, 69+72, s)-Nets in Base 3

(69, 69+72, 56)-Net over F₃ — Constructive and digital

Digital (69, 141, 56)-net over F₃, using

6 times m-reduction [i] based on digital (69, 147, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 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 (15, 93, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(69, 69+72, 82)-Net over F₃ — Digital

Digital (69, 141, 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+72, 493)-Net in Base 3 — Upper bound on s

There is no (69, 141, 494)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 831054 408069 393361 983735 116990 597740 303209 421304 481311 004396 682313 > 3¹⁴¹ [i]