MinT - Best Known (168, 168+61, s)-Nets in Base 3

Best Known (168, 168+61, s)-Nets in Base 3

(168, 168+61, 288)-Net over F₃ — Constructive and digital

Digital (168, 229, 288)-net over F₃, using

t-expansion [i] based on digital (167, 229, 288)-net over F₃, using
- 5 times m-reduction [i] based on digital (167, 234, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 78, 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]

(168, 168+61, 748)-Net over F₃ — Digital

Digital (168, 229, 748)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+61, 25431)-Net in Base 3 — Upper bound on s

There is no (168, 229, 25432)-net in base 3, because

1 times m-reduction [i] would yield (168, 228, 25432)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 077879 048789 639199 869690 845358 742413 793411 294956 259427 735063 815411 156052 338928 766454 982471 343068 095838 917265 > 3²²⁸ [i]