MinT - Best Known (198−71, 198, s)-Nets in Base 3

Best Known (198−71, 198, s)-Nets in Base 3

(198−71, 198, 156)-Net over F₃ — Constructive and digital

Digital (127, 198, 156)-net over F₃, using

12 times m-reduction [i] based on digital (127, 210, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 105, 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]

(198−71, 198, 253)-Net over F₃ — Digital

Digital (127, 198, 253)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−71, 198, 3336)-Net in Base 3 — Upper bound on s

There is no (127, 198, 3337)-net in base 3, because

1 times m-reduction [i] would yield (127, 197, 3337)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9884 775613 795009 764912 047831 330951 485334 136372 542527 418231 611793 438365 143465 030465 423808 947963 > 3¹⁹⁷ [i]