MinT - Best Known (224−81, 224, s)-Nets in Base 3

Best Known (224−81, 224, s)-Nets in Base 3

(224−81, 224, 156)-Net over F₃ — Constructive and digital

Digital (143, 224, 156)-net over F₃, using

18 times m-reduction [i] based on digital (143, 242, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 121, 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]

(224−81, 224, 275)-Net over F₃ — Digital

Digital (143, 224, 275)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−81, 224, 3564)-Net in Base 3 — Upper bound on s

There is no (143, 224, 3565)-net in base 3, because

1 times m-reduction [i] would yield (143, 223, 3565)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25263 298795 535407 716781 238116 529957 153955 923212 806526 647138 120534 041551 943322 675895 693490 848488 690500 048097 > 3²²³ [i]