MinT - Best Known (172−56, 172, s)-Nets in Base 3

Best Known (172−56, 172, s)-Nets in Base 3

(172−56, 172, 156)-Net over F₃ — Constructive and digital

Digital (116, 172, 156)-net over F₃, using

16 times m-reduction [i] based on digital (116, 188, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 94, 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]

(172−56, 172, 302)-Net over F₃ — Digital

Digital (116, 172, 302)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−56, 172, 4790)-Net in Base 3 — Upper bound on s

There is no (116, 172, 4791)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11643 034616 513077 618139 435155 409307 019170 251216 170979 758599 928275 683022 816317 904585 > 3¹⁷² [i]