MinT - Best Known (218−57, 218, s)-Nets in Base 3

Best Known (218−57, 218, s)-Nets in Base 3

(218−57, 218, 288)-Net over F₃ — Constructive and digital

Digital (161, 218, 288)-net over F₃, using

7 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]

(218−57, 218, 768)-Net over F₃ — Digital

Digital (161, 218, 768)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−57, 218, 28134)-Net in Base 3 — Upper bound on s

There is no (161, 218, 28135)-net in base 3, because

1 times m-reduction [i] would yield (161, 217, 28135)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 315311 422435 769979 598260 507370 357939 444422 438229 575465 905673 801187 660913 428846 196980 044990 242557 390793 > 3²¹⁷ [i]