MinT - Best Known (160−53, 160, s)-Nets in Base 3

Best Known (160−53, 160, s)-Nets in Base 3

(160−53, 160, 156)-Net over F₃ — Constructive and digital

Digital (107, 160, 156)-net over F₃, using

10 times m-reduction [i] based on digital (107, 170, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 85, 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]

(160−53, 160, 270)-Net over F₃ — Digital

Digital (107, 160, 270)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−53, 160, 4340)-Net in Base 3 — Upper bound on s

There is no (107, 160, 4341)-net in base 3, because

1 times m-reduction [i] would yield (107, 159, 4341)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7319 315152 485132 217545 564863 505617 141578 227264 780758 702960 114499 358284 784017 > 3¹⁵⁹ [i]