MinT - Best Known (216−83, 216, s)-Nets in Base 3

Best Known (216−83, 216, s)-Nets in Base 3

(216−83, 216, 156)-Net over F₃ — Constructive and digital

Digital (133, 216, 156)-net over F₃, using

6 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]

(216−83, 216, 224)-Net over F₃ — Digital

Digital (133, 216, 224)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−83, 216, 2523)-Net in Base 3 — Upper bound on s

There is no (133, 216, 2524)-net in base 3, because

1 times m-reduction [i] would yield (133, 215, 2524)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 846568 437167 718686 477358 788437 243209 088043 592401 361321 905579 971377 531055 139093 167052 716685 705244 341817 > 3²¹⁵ [i]