MinT - Best Known (214−81, 214, s)-Nets in Base 3

Best Known (214−81, 214, s)-Nets in Base 3

(214−81, 214, 156)-Net over F₃ — Constructive and digital

Digital (133, 214, 156)-net over F₃, using

8 times m-reduction [i] based on digital (133, 222, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 111, 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]

(214−81, 214, 232)-Net over F₃ — Digital

Digital (133, 214, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−81, 214, 2698)-Net in Base 3 — Upper bound on s

There is no (133, 214, 2699)-net in base 3, because

1 times m-reduction [i] would yield (133, 213, 2699)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 425174 583134 955545 552950 716708 457703 983337 923906 002339 986268 463557 313946 482247 503109 757908 455176 132657 > 3²¹³ [i]