MinT - Best Known (214−83, 214, s)-Nets in Base 3

Best Known (214−83, 214, s)-Nets in Base 3

(214−83, 214, 156)-Net over F₃ — Constructive and digital

Digital (131, 214, 156)-net over F₃, using

4 times m-reduction [i] based on digital (131, 218, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 109, 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]

(214−83, 214, 216)-Net over F₃ — Digital

Digital (131, 214, 216)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−83, 214, 2389)-Net in Base 3 — Upper bound on s

There is no (131, 214, 2390)-net in base 3, because

1 times m-reduction [i] would yield (131, 213, 2390)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 425868 922812 903794 565465 103573 668215 858376 190776 835526 591091 503735 019671 142867 880650 904763 235304 690653 > 3²¹³ [i]