MinT - Best Known (172−45, 172, s)-Nets in Base 3

Best Known (172−45, 172, s)-Nets in Base 3

(172−45, 172, 288)-Net over F₃ — Constructive and digital

Digital (127, 172, 288)-net over F₃, using

2 times m-reduction [i] based on digital (127, 174, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 58, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(172−45, 172, 630)-Net over F₃ — Digital

Digital (127, 172, 630)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−45, 172, 23118)-Net in Base 3 — Upper bound on s

There is no (127, 172, 23119)-net in base 3, because

1 times m-reduction [i] would yield (127, 171, 23119)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3873 380058 585452 050231 408380 732198 966420 814543 489885 209206 914031 976387 932480 294901 > 3¹⁷¹ [i]