MinT - Best Known (210−83, 210, s)-Nets in Base 3

Best Known (210−83, 210, s)-Nets in Base 3

(210−83, 210, 156)-Net over F₃ — Constructive and digital

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]

(210−83, 210, 202)-Net over F₃ — Digital

Digital (127, 210, 202)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−83, 210, 2142)-Net in Base 3 — Upper bound on s

There is no (127, 210, 2143)-net in base 3, because

1 times m-reduction [i] would yield (127, 209, 2143)-net in base 3, but
- 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]