MinT - Best Known (140−73, 140, s)-Nets in Base 3

Best Known (140−73, 140, s)-Nets in Base 3

(140−73, 140, 56)-Net over F₃ — Constructive and digital

Digital (67, 140, 56)-net over F₃, using

1 times m-reduction [i] based on digital (67, 141, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 52, 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, 89, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(140−73, 140, 72)-Net over F₃ — Digital

Digital (67, 140, 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]

(140−73, 140, 462)-Net in Base 3 — Upper bound on s

There is no (67, 140, 463)-net in base 3, because

1 times m-reduction [i] would yield (67, 139, 463)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 241493 559447 210792 189616 600042 848731 272202 046199 641819 757659 343801 > 3¹³⁹ [i]