MinT - Best Known (186−65, 186, s)-Nets in Base 3

Best Known (186−65, 186, s)-Nets in Base 3

(186−65, 186, 156)-Net over F₃ — Constructive and digital

Digital (121, 186, 156)-net over F₃, using

12 times m-reduction [i] based on digital (121, 198, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 99, 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]

(186−65, 186, 259)-Net over F₃ — Digital

Digital (121, 186, 259)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−65, 186, 3634)-Net in Base 3 — Upper bound on s

There is no (121, 186, 3635)-net in base 3, because

1 times m-reduction [i] would yield (121, 185, 3635)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18553 062166 011108 406362 338462 598578 492854 992722 000418 739335 392462 084835 346476 503852 825025 > 3¹⁸⁵ [i]