MinT - Best Known (211−77, 211, s)-Nets in Base 3

Best Known (211−77, 211, s)-Nets in Base 3

(211−77, 211, 156)-Net over F₃ — Constructive and digital

Digital (134, 211, 156)-net over F₃, using

13 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]

(211−77, 211, 255)-Net over F₃ — Digital

Digital (134, 211, 255)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−77, 211, 3217)-Net in Base 3 — Upper bound on s

There is no (134, 211, 3218)-net in base 3, because

1 times m-reduction [i] would yield (134, 210, 3218)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15819 153064 785174 472559 258328 469190 081604 252808 104374 746408 223954 596445 714540 179039 494299 346652 802013 > 3²¹⁰ [i]