MinT - Best Known (163−41, 163, s)-Nets in Base 3

Best Known (163−41, 163, s)-Nets in Base 3

(163−41, 163, 288)-Net over F₃ — Constructive and digital

Digital (122, 163, 288)-net over F₃, using

t-expansion [i] based on digital (121, 163, 288)-net over F₃, using
- 2 times m-reduction [i] based on digital (121, 165, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 55, 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]

(163−41, 163, 705)-Net over F₃ — Digital

Digital (122, 163, 705)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−41, 163, 30386)-Net in Base 3 — Upper bound on s

There is no (122, 163, 30387)-net in base 3, because

1 times m-reduction [i] would yield (122, 162, 30387)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 196699 137685 861343 383580 282886 269922 502368 960208 838443 093906 955216 964096 250809 > 3¹⁶² [i]