MinT - Best Known (231−84, 231, s)-Nets in Base 3

Best Known (231−84, 231, s)-Nets in Base 3

(231−84, 231, 156)-Net over F₃ — Constructive and digital

Digital (147, 231, 156)-net over F₃, using

19 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 125, 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]

(231−84, 231, 279)-Net over F₃ — Digital

Digital (147, 231, 279)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−84, 231, 3433)-Net in Base 3 — Upper bound on s

There is no (147, 231, 3434)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 164 903222 252574 263877 830008 345285 685395 304082 646503 552008 950880 237509 706188 286641 759106 732773 196841 199238 240469 > 3²³¹ [i]