MinT - Best Known (128−41, 128, s)-Nets in Base 3

Best Known (128−41, 128, s)-Nets in Base 3

(128−41, 128, 156)-Net over F₃ — Constructive and digital

Digital (87, 128, 156)-net over F₃, using

2 times m-reduction [i] based on digital (87, 130, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 65, 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]

(128−41, 128, 248)-Net over F₃ — Digital

Digital (87, 128, 248)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−41, 128, 4426)-Net in Base 3 — Upper bound on s

There is no (87, 128, 4427)-net in base 3, because

1 times m-reduction [i] would yield (87, 127, 4427)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 933430 586809 088518 559968 067574 545657 881914 443977 939891 622841 > 3¹²⁷ [i]