MinT - Best Known (71, 154, s)-Nets in Base 3

Best Known (71, 154, s)-Nets in Base 3

(71, 154, 56)-Net over F₃ — Constructive and digital

Digital (71, 154, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 56, 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, 98, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(71, 154, 84)-Net over F₃ — Digital

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]

(71, 154, 447)-Net in Base 3 — Upper bound on s

There is no (71, 154, 448)-net in base 3, because

1 times m-reduction [i] would yield (71, 153, 448)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 284884 947168 831657 114303 396768 765620 153571 849927 474002 519325 017508 089729 > 3¹⁵³ [i]