MinT - Best Known (235−75, 235, s)-Nets in Base 3

Best Known (235−75, 235, s)-Nets in Base 3

(235−75, 235, 172)-Net over F₃ — Constructive and digital

Digital (160, 235, 172)-net over F₃, using

3¹ times duplication [i] based on digital (159, 234, 172)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 50, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (109, 184, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 92, 74)-net over F₉, using
      - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(235−75, 235, 419)-Net over F₃ — Digital

Digital (160, 235, 419)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−75, 235, 7590)-Net in Base 3 — Upper bound on s

There is no (160, 235, 7591)-net in base 3, because

1 times m-reduction [i] would yield (160, 234, 7591)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4439 782380 316689 221384 497132 790902 891469 431221 299595 803748 291777 377507 289024 258439 110495 184896 992252 203166 626887 > 3²³⁴ [i]