MinT - Best Known (67, 67+63, s)-Nets in Base 3

Best Known (67, 67+63, s)-Nets in Base 3

(67, 67+63, 60)-Net over F₃ — Constructive and digital

Digital (67, 130, 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, 84, 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+63, 76)-Net over F₃ — Digital

Digital (67, 130, 76)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+63, 570)-Net in Base 3 — Upper bound on s

There is no (67, 130, 571)-net in base 3, because

1 times m-reduction [i] would yield (67, 129, 571)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 234543 411653 067365 202689 767946 672562 260070 737355 435390 545331 > 3¹²⁹ [i]