MinT - Best Known (218−81, 218, s)-Nets in Base 3

Best Known (218−81, 218, s)-Nets in Base 3

(218−81, 218, 156)-Net over F₃ — Constructive and digital

Digital (137, 218, 156)-net over F₃, using

12 times m-reduction [i] based on digital (137, 230, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 115, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(218−81, 218, 249)-Net over F₃ — Digital

Digital (137, 218, 249)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−81, 218, 3016)-Net in Base 3 — Upper bound on s

There is no (137, 218, 3017)-net in base 3, because

1 times m-reduction [i] would yield (137, 217, 3017)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 460997 733967 108245 401793 354342 493735 945409 070442 367408 797743 071911 350350 699623 780396 443671 535633 381665 > 3²¹⁷ [i]