MinT - Best Known (133−65, 133, s)-Nets in Base 3

Best Known (133−65, 133, s)-Nets in Base 3

(133−65, 133, 60)-Net over F₃ — Constructive and digital

Digital (68, 133, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 47, 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 (21, 86, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]

(133−65, 133, 76)-Net over F₃ — Digital

Digital (68, 133, 76)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−65, 133, 563)-Net in Base 3 — Upper bound on s

There is no (68, 133, 564)-net in base 3, because

1 times m-reduction [i] would yield (68, 132, 564)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 984 453150 984977 257723 056691 339445 416661 584614 128151 005450 075393 > 3¹³² [i]