MinT - Best Known (213−57, 213, s)-Nets in Base 3

Best Known (213−57, 213, s)-Nets in Base 3

(213−57, 213, 288)-Net over F₃ — Constructive and digital

Digital (156, 213, 288)-net over F₃, using

t-expansion [i] based on digital (155, 213, 288)-net over F₃, using
- 3 times m-reduction [i] based on digital (155, 216, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 72, 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]

(213−57, 213, 691)-Net over F₃ — Digital

Digital (156, 213, 691)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−57, 213, 23117)-Net in Base 3 — Upper bound on s

There is no (156, 213, 23118)-net in base 3, because

1 times m-reduction [i] would yield (156, 212, 23118)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141174 338957 257920 026194 413246 625364 759961 952400 361384 048612 405565 233260 285415 911053 543365 874435 628921 > 3²¹² [i]