MinT - Best Known (219−73, 219, s)-Nets in Base 3

Best Known (219−73, 219, s)-Nets in Base 3

(219−73, 219, 162)-Net over F₃ — Constructive and digital

Digital (146, 219, 162)-net over F₃, using

9 times m-reduction [i] based on digital (146, 228, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 114, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(219−73, 219, 344)-Net over F₃ — Digital

Digital (146, 219, 344)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−73, 219, 5497)-Net in Base 3 — Upper bound on s

There is no (146, 219, 5498)-net in base 3, because

1 times m-reduction [i] would yield (146, 218, 5498)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 341264 357961 341968 262757 847902 533832 322570 427241 066553 401122 604290 410191 139493 580442 025996 628301 735657 > 3²¹⁸ [i]