MinT - Best Known (134−67, 134, s)-Nets in Base 3

Best Known (134−67, 134, s)-Nets in Base 3

(134−67, 134, 56)-Net over F₃ — Constructive and digital

Digital (67, 134, 56)-net over F₃, using

7 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)

(134−67, 134, 72)-Net over F₃ — Digital

Digital (67, 134, 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]

(134−67, 134, 519)-Net in Base 3 — Upper bound on s

There is no (67, 134, 520)-net in base 3, because

1 times m-reduction [i] would yield (67, 133, 520)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2965 962850 738619 110871 859587 188827 817302 336282 980968 284634 222097 > 3¹³³ [i]