MinT - Best Known (42, 81, s)-Nets in Base 3

Best Known (42, 81, s)-Nets in Base 3

(42, 81, 44)-Net over F₃ — Constructive and digital

Digital (42, 81, 44)-net over F₃, using

1 times m-reduction [i] based on digital (42, 82, 44)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 27, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 55, 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]

(42, 81, 56)-Net over F₃ — Digital

Digital (42, 81, 56)-net over F₃, using

t-expansion [i] based on digital (40, 81, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(42, 81, 386)-Net in Base 3 — Upper bound on s

There is no (42, 81, 387)-net in base 3, because

1 times m-reduction [i] would yield (42, 80, 387)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 151 104248 295856 861820 588949 568968 962187 > 3⁸⁰ [i]