MinT - Best Known (213−84, 213, s)-Nets in Base 3

Best Known (213−84, 213, s)-Nets in Base 3

(213−84, 213, 156)-Net over F₃ — Constructive and digital

Digital (129, 213, 156)-net over F₃, using

1 times m-reduction [i] based on digital (129, 214, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 107, 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]

(213−84, 213, 205)-Net over F₃ — Digital

Digital (129, 213, 205)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−84, 213, 2128)-Net in Base 3 — Upper bound on s

There is no (129, 213, 2129)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 424949 851964 057090 549629 033559 960251 372479 400255 324117 425899 835626 066554 353772 998800 144141 976894 435657 > 3²¹³ [i]