MinT - Best Known (127, 127+82, s)-Nets in Base 3

Best Known (127, 127+82, s)-Nets in Base 3

(127, 127+82, 156)-Net over F₃ — Constructive and digital

Digital (127, 209, 156)-net over F₃, using

1 times m-reduction [i] based on digital (127, 210, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 105, 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]

(127, 127+82, 205)-Net over F₃ — Digital

Digital (127, 209, 205)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+82, 2142)-Net in Base 3 — Upper bound on s

There is no (127, 209, 2143)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5254 724633 148394 581790 621572 749268 281688 744164 482670 752814 081727 772082 426035 604484 028795 874636 534895 > 3²⁰⁹ [i]