MinT - Best Known (170−54, 170, s)-Nets in Base 3

Best Known (170−54, 170, s)-Nets in Base 3

(170−54, 170, 156)-Net over F₃ — Constructive and digital

Digital (116, 170, 156)-net over F₃, using

18 times m-reduction [i] based on digital (116, 188, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 94, 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]

(170−54, 170, 322)-Net over F₃ — Digital

Digital (116, 170, 322)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−54, 170, 5487)-Net in Base 3 — Upper bound on s

There is no (116, 170, 5488)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1292 349558 739138 027069 980056 001677 020513 760726 035538 475709 636015 111052 549922 380737 > 3¹⁷⁰ [i]