MinT - Best Known (133−63, 133, s)-Nets in Base 3

Best Known (133−63, 133, s)-Nets in Base 3

(133−63, 133, 60)-Net over F₃ — Constructive and digital

Digital (70, 133, 60)-net over F₃, using

5 times m-reduction [i] based on digital (70, 138, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 49, 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, 89, 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]

(133−63, 133, 83)-Net over F₃ — Digital

Digital (70, 133, 83)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−63, 133, 637)-Net in Base 3 — Upper bound on s

There is no (70, 133, 638)-net in base 3, because

1 times m-reduction [i] would yield (70, 132, 638)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 962 979547 634314 167901 047454 759459 295094 798231 949549 202559 743177 > 3¹³² [i]