MinT - Best Known (223−89, 223, s)-Nets in Base 3

Best Known (223−89, 223, s)-Nets in Base 3

(223−89, 223, 156)-Net over F₃ — Constructive and digital

Digital (134, 223, 156)-net over F₃, using

1 times m-reduction [i] based on digital (134, 224, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 112, 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]

(223−89, 223, 207)-Net over F₃ — Digital

Digital (134, 223, 207)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−89, 223, 2160)-Net in Base 3 — Upper bound on s

There is no (134, 223, 2161)-net in base 3, because

1 times m-reduction [i] would yield (134, 222, 2161)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8380 930275 275648 653283 339394 955830 457161 889234 869891 016847 805501 223996 112502 424953 470816 149603 103063 877105 > 3²²² [i]