MinT - Best Known (170−55, 170, s)-Nets in Base 3

Best Known (170−55, 170, s)-Nets in Base 3

(170−55, 170, 156)-Net over F₃ — Constructive and digital

Digital (115, 170, 156)-net over F₃, using

16 times m-reduction [i] based on digital (115, 186, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 93, 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]

(170−55, 170, 305)-Net over F₃ — Digital

Digital (115, 170, 305)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−55, 170, 5267)-Net in Base 3 — Upper bound on s

There is no (115, 170, 5268)-net in base 3, because

1 times m-reduction [i] would yield (115, 169, 5268)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 430 505146 400890 097108 102224 848917 561028 178297 874736 788713 049836 190766 256704 774321 > 3¹⁶⁹ [i]