MinT - Best Known (164−53, 164, s)-Nets in Base 3

Best Known (164−53, 164, s)-Nets in Base 3

(164−53, 164, 156)-Net over F₃ — Constructive and digital

Digital (111, 164, 156)-net over F₃, using

14 times m-reduction [i] based on digital (111, 178, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 89, 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]

(164−53, 164, 297)-Net over F₃ — Digital

Digital (111, 164, 297)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−53, 164, 5144)-Net in Base 3 — Upper bound on s

There is no (111, 164, 5145)-net in base 3, because

1 times m-reduction [i] would yield (111, 163, 5145)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 592778 506402 247210 723263 295612 011358 039398 852260 789453 875470 413776 852410 762713 > 3¹⁶³ [i]