MinT - Best Known (131−68, 131, s)-Nets in Base 3

Best Known (131−68, 131, s)-Nets in Base 3

(131−68, 131, 52)-Net over F₃ — Constructive and digital

Digital (63, 131, 52)-net over F₃, using

2 times m-reduction [i] based on digital (63, 133, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 48, 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, 85, 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]

(131−68, 131, 64)-Net over F₃ — Digital

Digital (63, 131, 64)-net over F₃, using

t-expansion [i] based on digital (49, 131, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(131−68, 131, 433)-Net in Base 3 — Upper bound on s

There is no (63, 131, 434)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 321 272273 235375 866082 571074 580659 341621 084197 100238 536977 523445 > 3¹³¹ [i]