MinT - Best Known (172−62, 172, s)-Nets in Base 3

Best Known (172−62, 172, s)-Nets in Base 3

(172−62, 172, 156)-Net over F₃ — Constructive and digital

Digital (110, 172, 156)-net over F₃, using

4 times m-reduction [i] based on digital (110, 176, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 88, 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]

(172−62, 172, 221)-Net over F₃ — Digital

Digital (110, 172, 221)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−62, 172, 2725)-Net in Base 3 — Upper bound on s

There is no (110, 172, 2726)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11676 734957 575446 733555 896301 291381 356099 948553 478826 265730 966369 623962 110078 987689 > 3¹⁷² [i]