MinT - Best Known (129−41, 129, s)-Nets in Base 3

Best Known (129−41, 129, s)-Nets in Base 3

(129−41, 129, 156)-Net over F₃ — Constructive and digital

Digital (88, 129, 156)-net over F₃, using

3 times m-reduction [i] based on digital (88, 132, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 66, 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]

(129−41, 129, 255)-Net over F₃ — Digital

Digital (88, 129, 255)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−41, 129, 4677)-Net in Base 3 — Upper bound on s

There is no (88, 129, 4678)-net in base 3, because

1 times m-reduction [i] would yield (88, 128, 4678)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 796375 232047 582995 874992 595797 342083 549228 284020 850305 557161 > 3¹²⁸ [i]