MinT - Best Known (73, 73+81, s)-Nets in Base 3

Best Known (73, 73+81, s)-Nets in Base 3

(73, 73+81, 56)-Net over F₃ — Constructive and digital

Digital (73, 154, 56)-net over F₃, using

5 times m-reduction [i] based on digital (73, 159, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 58, 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, 101, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(73, 73+81, 84)-Net over F₃ — Digital

Digital (73, 154, 84)-net over F₃, using

t-expansion [i] based on digital (71, 154, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(73, 73+81, 488)-Net in Base 3 — Upper bound on s

There is no (73, 154, 489)-net in base 3, because

1 times m-reduction [i] would yield (73, 153, 489)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 287854 983124 041580 918217 500656 958910 466947 939631 199819 400691 333260 020001 > 3¹⁵³ [i]