MinT - Best Known (210−57, 210, s)-Nets in Base 3

Best Known (210−57, 210, s)-Nets in Base 3

(210−57, 210, 288)-Net over F₃ — Constructive and digital

Digital (153, 210, 288)-net over F₃, using

3 times m-reduction [i] based on digital (153, 213, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 71, 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]

(210−57, 210, 649)-Net over F₃ — Digital

Digital (153, 210, 649)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−57, 210, 20547)-Net in Base 3 — Upper bound on s

There is no (153, 210, 20548)-net in base 3, because

1 times m-reduction [i] would yield (153, 209, 20548)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5229 993127 852369 814252 237791 089941 224200 589796 737468 884266 588867 002081 750073 668566 248661 241558 930305 > 3²⁰⁹ [i]