MinT - Best Known (116, 116+59, s)-Nets in Base 3

Best Known (116, 116+59, s)-Nets in Base 3

(116, 116+59, 156)-Net over F₃ — Constructive and digital

Digital (116, 175, 156)-net over F₃, using

13 times m-reduction [i] based on digital (116, 188, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 94, 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]

(116, 116+59, 275)-Net over F₃ — Digital

Digital (116, 175, 275)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(116, 116+59, 4225)-Net in Base 3 — Upper bound on s

There is no (116, 175, 4226)-net in base 3, because

1 times m-reduction [i] would yield (116, 174, 4226)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104641 275641 565226 112061 236786 910722 843274 153314 265613 778237 033493 708599 401135 115181 > 3¹⁷⁴ [i]