MinT - Best Known (136, 136+79, s)-Nets in Base 3

Best Known (136, 136+79, s)-Nets in Base 3

(136, 136+79, 156)-Net over F₃ — Constructive and digital

Digital (136, 215, 156)-net over F₃, using

13 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]

(136, 136+79, 254)-Net over F₃ — Digital

Digital (136, 215, 254)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+79, 3156)-Net in Base 3 — Upper bound on s

There is no (136, 215, 3157)-net in base 3, because

1 times m-reduction [i] would yield (136, 214, 3157)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 278141 295565 688740 895450 891582 036703 881884 952373 903894 088569 308842 765201 634721 958507 920076 216312 195995 > 3²¹⁴ [i]