MinT - Best Known (165−55, 165, s)-Nets in Base 3

Best Known (165−55, 165, s)-Nets in Base 3

(165−55, 165, 156)-Net over F₃ — Constructive and digital

Digital (110, 165, 156)-net over F₃, using

11 times m-reduction [i] based on digital (110, 176, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 88, 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]

(165−55, 165, 271)-Net over F₃ — Digital

Digital (110, 165, 271)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−55, 165, 4293)-Net in Base 3 — Upper bound on s

There is no (110, 165, 4294)-net in base 3, because

1 times m-reduction [i] would yield (110, 164, 4294)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 778531 477908 838980 729621 230447 022692 164389 825298 910919 728242 442192 269497 614705 > 3¹⁶⁴ [i]