MinT - Best Known (197−91, 197, s)-Nets in Base 3

Best Known (197−91, 197, s)-Nets in Base 3

(197−91, 197, 76)-Net over F₃ — Constructive and digital

Digital (106, 197, 76)-net over F₃, using

1 times m-reduction [i] based on digital (106, 198, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 61, 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 (45, 137, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(197−91, 197, 123)-Net over F₃ — Digital

Digital (106, 197, 123)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−91, 197, 1011)-Net in Base 3 — Upper bound on s

There is no (106, 197, 1012)-net in base 3, because

1 times m-reduction [i] would yield (106, 196, 1012)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3376 853285 376966 766200 587673 599219 367591 889684 473367 616613 601746 418463 415497 278871 621257 459017 > 3¹⁹⁶ [i]