MinT - Best Known (193−69, 193, s)-Nets in Base 3

Best Known (193−69, 193, s)-Nets in Base 3

(193−69, 193, 156)-Net over F₃ — Constructive and digital

Digital (124, 193, 156)-net over F₃, using

11 times m-reduction [i] based on digital (124, 204, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 102, 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]

(193−69, 193, 250)-Net over F₃ — Digital

Digital (124, 193, 250)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−69, 193, 3314)-Net in Base 3 — Upper bound on s

There is no (124, 193, 3315)-net in base 3, because

1 times m-reduction [i] would yield (124, 192, 3315)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 658517 780718 232076 080678 213333 298329 777455 014135 112376 514138 637831 870452 603961 374173 654469 > 3¹⁹² [i]