MinT - Best Known (67, 67+61, s)-Nets in Base 3

Best Known (67, 67+61, s)-Nets in Base 3

(67, 67+61, 60)-Net over F₃ — Constructive and digital

Digital (67, 128, 60)-net over F₃, using

1 times m-reduction [i] based on digital (67, 129, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 46, 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, 83, 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]

(67, 67+61, 79)-Net over F₃ — Digital

Digital (67, 128, 79)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+61, 601)-Net in Base 3 — Upper bound on s

There is no (67, 128, 602)-net in base 3, because

1 times m-reduction [i] would yield (67, 127, 602)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 044307 197801 036752 187877 216509 828679 262637 212058 223851 617213 > 3¹²⁷ [i]