MinT - Best Known (234−87, 234, s)-Nets in Base 3

Best Known (234−87, 234, s)-Nets in Base 3

(234−87, 234, 156)-Net over F₃ — Constructive and digital

Digital (147, 234, 156)-net over F₃, using

16 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 125, 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]

(234−87, 234, 265)-Net over F₃ — Digital

Digital (147, 234, 265)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−87, 234, 3206)-Net in Base 3 — Upper bound on s

There is no (147, 234, 3207)-net in base 3, because

1 times m-reduction [i] would yield (147, 233, 3207)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1478 870987 898210 877567 263870 778080 567732 186087 672702 911315 776266 629818 710109 106519 901800 916651 507192 848988 155979 > 3²³³ [i]