MinT - Best Known (168−55, 168, s)-Nets in Base 3

Best Known (168−55, 168, s)-Nets in Base 3

(168−55, 168, 156)-Net over F₃ — Constructive and digital

Digital (113, 168, 156)-net over F₃, using

14 times m-reduction [i] based on digital (113, 182, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 91, 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]

(168−55, 168, 291)-Net over F₃ — Digital

Digital (113, 168, 291)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−55, 168, 4854)-Net in Base 3 — Upper bound on s

There is no (113, 168, 4855)-net in base 3, because

1 times m-reduction [i] would yield (113, 167, 4855)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48 042100 717591 537937 997315 156146 909313 679011 408434 239054 030745 746337 406009 593131 > 3¹⁶⁷ [i]