MinT - Best Known (163−57, 163, s)-Nets in Base 3

Best Known (163−57, 163, s)-Nets in Base 3

(163−57, 163, 156)-Net over F₃ — Constructive and digital

Digital (106, 163, 156)-net over F₃, using

5 times m-reduction [i] based on digital (106, 168, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 84, 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]

(163−57, 163, 231)-Net over F₃ — Digital

Digital (106, 163, 231)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−57, 163, 3226)-Net in Base 3 — Upper bound on s

There is no (106, 163, 3227)-net in base 3, because

1 times m-reduction [i] would yield (106, 162, 3227)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 196742 210296 063441 744289 277104 130374 730877 834161 065600 718499 958081 869126 258601 > 3¹⁶² [i]