MinT - Best Known (165−91, 165, s)-Nets in Base 3

Best Known (165−91, 165, s)-Nets in Base 3

(165−91, 165, 52)-Net over F₃ — Constructive and digital

Digital (74, 165, 52)-net over F₃, using

1 times m-reduction [i] based on digital (74, 166, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 59, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (15, 107, 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]

(165−91, 165, 84)-Net over F₃ — Digital

Digital (74, 165, 84)-net over F₃, using

t-expansion [i] based on digital (71, 165, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(165−91, 165, 440)-Net in Base 3 — Upper bound on s

There is no (74, 165, 441)-net in base 3, because

1 times m-reduction [i] would yield (74, 164, 441)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 912131 663245 472408 103181 509633 658672 568912 368449 520857 235165 980710 190910 820291 > 3¹⁶⁴ [i]