MinT - Best Known (67, 138, s)-Nets in Base 3

Best Known (67, 138, s)-Nets in Base 3

(67, 138, 56)-Net over F₃ — Constructive and digital

Digital (67, 138, 56)-net over F₃, using

3 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)

(67, 138, 72)-Net over F₃ — Digital

Digital (67, 138, 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]

(67, 138, 479)-Net in Base 3 — Upper bound on s

There is no (67, 138, 480)-net in base 3, because

1 times m-reduction [i] would yield (67, 137, 480)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 247006 230128 151561 238379 247131 359533 275997 937347 458632 404501 676929 > 3¹³⁷ [i]