MinT - Best Known (222−86, 222, s)-Nets in Base 3

Best Known (222−86, 222, s)-Nets in Base 3

(222−86, 222, 156)-Net over F₃ — Constructive and digital

Digital (136, 222, 156)-net over F₃, using

6 times m-reduction [i] based on digital (136, 228, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 114, 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]

(222−86, 222, 224)-Net over F₃ — Digital

Digital (136, 222, 224)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−86, 222, 2410)-Net in Base 3 — Upper bound on s

There is no (136, 222, 2411)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8348 847232 580196 576053 284321 285901 630749 529804 889317 308261 785732 259370 588278 082521 485010 424393 525235 428475 > 3²²² [i]