MinT - Best Known (68, 68+61, s)-Nets in Base 3

Best Known (68, 68+61, s)-Nets in Base 3

(68, 68+61, 60)-Net over F₃ — Constructive and digital

Digital (68, 129, 60)-net over F₃, using

3 times m-reduction [i] based on digital (68, 132, 60)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 47, 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, 85, 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]

(68, 68+61, 81)-Net over F₃ — Digital

Digital (68, 129, 81)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+61, 624)-Net in Base 3 — Upper bound on s

There is no (68, 129, 625)-net in base 3, because

1 times m-reduction [i] would yield (68, 128, 625)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 850187 054561 523332 559534 792285 376235 477426 484930 070168 971801 > 3¹²⁸ [i]