MinT - Best Known (188−63, 188, s)-Nets in Base 3

Best Known (188−63, 188, s)-Nets in Base 3

(188−63, 188, 156)-Net over F₃ — Constructive and digital

Digital (125, 188, 156)-net over F₃, using

18 times m-reduction [i] based on digital (125, 206, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 103, 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]

(188−63, 188, 297)-Net over F₃ — Digital

Digital (125, 188, 297)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−63, 188, 4659)-Net in Base 3 — Upper bound on s

There is no (125, 188, 4660)-net in base 3, because

1 times m-reduction [i] would yield (125, 187, 4660)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 167673 497317 262192 522007 133696 630011 747508 061297 669949 031247 796900 128156 135574 441859 176945 > 3¹⁸⁷ [i]