MinT - Best Known (171−57, 171, s)-Nets in Base 3

Best Known (171−57, 171, s)-Nets in Base 3

(171−57, 171, 156)-Net over F₃ — Constructive and digital

Digital (114, 171, 156)-net over F₃, using

13 times m-reduction [i] based on digital (114, 184, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 92, 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]

(171−57, 171, 279)-Net over F₃ — Digital

Digital (114, 171, 279)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−57, 171, 4426)-Net in Base 3 — Upper bound on s

There is no (114, 171, 4427)-net in base 3, because

1 times m-reduction [i] would yield (114, 170, 4427)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1290 853517 431582 840743 794236 376369 959454 949880 455331 049812 841136 931306 960851 426217 > 3¹⁷⁰ [i]