MinT - Best Known (197−75, 197, s)-Nets in Base 3

Best Known (197−75, 197, s)-Nets in Base 3

(197−75, 197, 156)-Net over F₃ — Constructive and digital

Digital (122, 197, 156)-net over F₃, using

3 times m-reduction [i] based on digital (122, 200, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 100, 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]

(197−75, 197, 213)-Net over F₃ — Digital

Digital (122, 197, 213)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−75, 197, 2431)-Net in Base 3 — Upper bound on s

There is no (122, 197, 2432)-net in base 3, because

1 times m-reduction [i] would yield (122, 196, 2432)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3295 667485 139164 797419 854506 778073 336511 919826 530378 720664 157380 860488 738896 658276 913801 515777 > 3¹⁹⁶ [i]