MinT - Best Known (216−85, 216, s)-Nets in Base 3

Best Known (216−85, 216, s)-Nets in Base 3

(216−85, 216, 156)-Net over F₃ — Constructive and digital

Digital (131, 216, 156)-net over F₃, using

2 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]

(216−85, 216, 209)-Net over F₃ — Digital

Digital (131, 216, 209)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−85, 216, 2245)-Net in Base 3 — Upper bound on s

There is no (131, 216, 2246)-net in base 3, because

1 times m-reduction [i] would yield (131, 215, 2246)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 857146 436452 062234 400174 583267 655329 783222 681826 514148 336332 877394 530742 765598 515257 529974 797913 806269 > 3²¹⁵ [i]