MinT - Best Known (218−59, 218, s)-Nets in Base 3

Best Known (218−59, 218, s)-Nets in Base 3

(218−59, 218, 288)-Net over F₃ — Constructive and digital

Digital (159, 218, 288)-net over F₃, using

4 times m-reduction [i] based on digital (159, 222, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 74, 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]

(218−59, 218, 677)-Net over F₃ — Digital

Digital (159, 218, 677)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−59, 218, 21662)-Net in Base 3 — Upper bound on s

There is no (159, 218, 21663)-net in base 3, because

1 times m-reduction [i] would yield (159, 217, 21663)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 333647 448504 592107 028871 873487 944495 994606 146900 768759 504488 079595 581546 991212 691918 539405 348687 490055 > 3²¹⁷ [i]