MinT - Best Known (164, 164+61, s)-Nets in Base 3

Best Known (164, 164+61, s)-Nets in Base 3

(164, 164+61, 288)-Net over F₃ — Constructive and digital

Digital (164, 225, 288)-net over F₃, using

t-expansion [i] based on digital (163, 225, 288)-net over F₃, using
- 3 times m-reduction [i] based on digital (163, 228, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 76, 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]

(164, 164+61, 691)-Net over F₃ — Digital

Digital (164, 225, 691)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+61, 21962)-Net in Base 3 — Upper bound on s

There is no (164, 225, 21963)-net in base 3, because

1 times m-reduction [i] would yield (164, 224, 21963)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75070 278034 098225 390123 266730 132904 373361 355174 345780 459312 277137 184666 711472 121725 855930 327824 824369 307133 > 3²²⁴ [i]