MinT - Best Known (224−83, 224, s)-Nets in Base 3

Best Known (224−83, 224, s)-Nets in Base 3

(224−83, 224, 156)-Net over F₃ — Constructive and digital

Digital (141, 224, 156)-net over F₃, using

14 times m-reduction [i] based on digital (141, 238, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 119, 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]

(224−83, 224, 257)-Net over F₃ — Digital

Digital (141, 224, 257)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−83, 224, 3136)-Net in Base 3 — Upper bound on s

There is no (141, 224, 3137)-net in base 3, because

1 times m-reduction [i] would yield (141, 223, 3137)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25241 165956 806777 324956 942561 479548 616177 706467 795444 776226 742934 071194 229275 130724 987879 968483 567169 676515 > 3²²³ [i]