MinT - Best Known (68, 68+71, s)-Nets in Base 3

Best Known (68, 68+71, s)-Nets in Base 3

(68, 68+71, 56)-Net over F₃ — Constructive and digital

Digital (68, 139, 56)-net over F₃, using

5 times m-reduction [i] based on digital (68, 144, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 53, 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, 91, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(68, 68+71, 72)-Net over F₃ — Digital

Digital (68, 139, 72)-net over F₃, using

t-expansion [i] based on digital (67, 139, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 68+71, 495)-Net in Base 3 — Upper bound on s

There is no (68, 139, 496)-net in base 3, because

1 times m-reduction [i] would yield (68, 138, 496)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 724142 875171 939212 927067 612089 251221 287414 361206 669527 394781 031361 > 3¹³⁸ [i]