MinT - Best Known (165−56, 165, s)-Nets in Base 3

Best Known (165−56, 165, s)-Nets in Base 3

(165−56, 165, 156)-Net over F₃ — Constructive and digital

Digital (109, 165, 156)-net over F₃, using

9 times m-reduction [i] based on digital (109, 174, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 87, 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]

(165−56, 165, 256)-Net over F₃ — Digital

Digital (109, 165, 256)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−56, 165, 3633)-Net in Base 3 — Upper bound on s

There is no (109, 165, 3634)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 331710 933425 702211 272911 673237 468874 407494 748717 426042 260317 323575 927840 216345 > 3¹⁶⁵ [i]