MinT - Best Known (198−77, 198, s)-Nets in Base 3

Best Known (198−77, 198, s)-Nets in Base 3

(198−77, 198, 156)-Net over F₃ — Constructive and digital

Digital (121, 198, 156)-net over F₃, using

trace code for nets [i] based on digital (22, 99, 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]

(198−77, 198, 201)-Net over F₃ — Digital

Digital (121, 198, 201)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−77, 198, 2197)-Net in Base 3 — Upper bound on s

There is no (121, 198, 2198)-net in base 3, because

1 times m-reduction [i] would yield (121, 197, 2198)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9884 059345 812610 410359 569646 644513 134951 148833 937974 115028 738348 850255 156698 026009 647017 107317 > 3¹⁹⁷ [i]