MinT - Best Known (163−51, 163, s)-Nets in Base 3

Best Known (163−51, 163, s)-Nets in Base 3

(163−51, 163, 156)-Net over F₃ — Constructive and digital

Digital (112, 163, 156)-net over F₃, using

17 times m-reduction [i] based on digital (112, 180, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 90, 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]

(163−51, 163, 325)-Net over F₃ — Digital

Digital (112, 163, 325)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−51, 163, 6261)-Net in Base 3 — Upper bound on s

There is no (112, 163, 6262)-net in base 3, because

1 times m-reduction [i] would yield (112, 162, 6262)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 197359 666345 228669 895918 489970 106738 901295 466997 356066 124455 736467 917388 831293 > 3¹⁶² [i]