MinT - Best Known (219−78, 219, s)-Nets in Base 3

Best Known (219−78, 219, s)-Nets in Base 3

(219−78, 219, 156)-Net over F₃ — Constructive and digital

Digital (141, 219, 156)-net over F₃, using

19 times m-reduction [i] based on digital (141, 238, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 119, 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]

(219−78, 219, 282)-Net over F₃ — Digital

Digital (141, 219, 282)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−78, 219, 3639)-Net in Base 3 — Upper bound on s

There is no (141, 219, 3640)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 309 723425 846375 519645 721343 113329 031332 856514 788532 251509 454025 509576 374134 423118 890701 249112 066768 800161 > 3²¹⁹ [i]