MinT - Best Known (216−77, 216, s)-Nets in Base 3

Best Known (216−77, 216, s)-Nets in Base 3

(216−77, 216, 156)-Net over F₃ — Constructive and digital

Digital (139, 216, 156)-net over F₃, using

18 times m-reduction [i] based on digital (139, 234, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 117, 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]

(216−77, 216, 278)-Net over F₃ — Digital

Digital (139, 216, 278)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−77, 216, 3723)-Net in Base 3 — Upper bound on s

There is no (139, 216, 3724)-net in base 3, because

1 times m-reduction [i] would yield (139, 215, 3724)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 834057 410578 741944 000759 537192 696358 542179 313673 030958 926001 294728 916867 620876 738696 209950 610691 535625 > 3²¹⁵ [i]