MinT - Best Known (227−61, 227, s)-Nets in Base 3

Best Known (227−61, 227, s)-Nets in Base 3

(227−61, 227, 288)-Net over F₃ — Constructive and digital

Digital (166, 227, 288)-net over F₃, using

t-expansion [i] based on digital (165, 227, 288)-net over F₃, using
- 4 times m-reduction [i] based on digital (165, 231, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 77, 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]

(227−61, 227, 719)-Net over F₃ — Digital

Digital (166, 227, 719)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−61, 227, 23633)-Net in Base 3 — Upper bound on s

There is no (166, 227, 23634)-net in base 3, because

1 times m-reduction [i] would yield (166, 226, 23634)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 675461 728472 109139 043812 508988 311155 715853 652572 666187 366918 387315 072207 379601 993723 417919 274007 331080 993805 > 3²²⁶ [i]