MinT - Best Known (219−57, 219, s)-Nets in Base 3

Best Known (219−57, 219, s)-Nets in Base 3

(219−57, 219, 288)-Net over F₃ — Constructive and digital

Digital (162, 219, 288)-net over F₃, using

t-expansion [i] based on digital (161, 219, 288)-net over F₃, using
- 6 times m-reduction [i] based on digital (161, 225, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 75, 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]

(219−57, 219, 784)-Net over F₃ — Digital

Digital (162, 219, 784)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−57, 219, 29261)-Net in Base 3 — Upper bound on s

There is no (162, 219, 29262)-net in base 3, because

1 times m-reduction [i] would yield (162, 218, 29262)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 950722 634775 743542 109554 493665 132787 521522 474740 849221 965842 388595 460828 425817 227469 582024 143206 883193 > 3²¹⁸ [i]