MinT - Best Known (141, 141+65, s)-Nets in Base 3

Best Known (141, 141+65, s)-Nets in Base 3

(141, 141+65, 167)-Net over F₃ — Constructive and digital

Digital (141, 206, 167)-net over F₃, using

3¹ times duplication [i] based on digital (140, 205, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 41, 19)-net over F₃, using
    - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
        an explicitly constructive algebraic function field [i]
  2. digital (99, 164, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 82, 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]

(141, 141+65, 389)-Net over F₃ — Digital

Digital (141, 206, 389)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(141, 141+65, 7253)-Net in Base 3 — Upper bound on s

There is no (141, 206, 7254)-net in base 3, because

1 times m-reduction [i] would yield (141, 205, 7254)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64 770823 655705 393376 893929 659014 665492 775442 957600 688846 599241 382504 995909 285864 380583 933387 892033 > 3²⁰⁵ [i]