MinT - Best Known (169−55, 169, s)-Nets in Base 3

Best Known (169−55, 169, s)-Nets in Base 3

(169−55, 169, 156)-Net over F₃ — Constructive and digital

Digital (114, 169, 156)-net over F₃, using

15 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]

(169−55, 169, 298)-Net over F₃ — Digital

Digital (114, 169, 298)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−55, 169, 5056)-Net in Base 3 — Upper bound on s

There is no (114, 169, 5057)-net in base 3, because

1 times m-reduction [i] would yield (114, 168, 5057)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 143 578898 523792 237544 141451 384202 382699 509284 965240 864893 119273 676224 462949 959867 > 3¹⁶⁸ [i]