MinT - Best Known (180−65, 180, s)-Nets in Base 3

Best Known (180−65, 180, s)-Nets in Base 3

(180−65, 180, 156)-Net over F₃ — Constructive and digital

Digital (115, 180, 156)-net over F₃, using

6 times m-reduction [i] based on digital (115, 186, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 93, 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]

(180−65, 180, 229)-Net over F₃ — Digital

Digital (115, 180, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−65, 180, 2952)-Net in Base 3 — Upper bound on s

There is no (115, 180, 2953)-net in base 3, because

1 times m-reduction [i] would yield (115, 179, 2953)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 587960 337903 535688 333554 419501 679951 842232 801776 138583 689528 686047 229453 166180 615809 > 3¹⁷⁹ [i]