MinT - Best Known (166−41, 166, s)-Nets in Base 3

Best Known (166−41, 166, s)-Nets in Base 3

(166−41, 166, 328)-Net over F₃ — Constructive and digital

Digital (125, 166, 328)-net over F₃, using

3² times duplication [i] based on digital (123, 164, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 41, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(166−41, 166, 770)-Net over F₃ — Digital

Digital (125, 166, 770)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−41, 166, 35833)-Net in Base 3 — Upper bound on s

There is no (125, 166, 35834)-net in base 3, because

1 times m-reduction [i] would yield (125, 165, 35834)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 310132 709700 164421 894393 746618 406681 756937 913020 696726 231691 779301 342558 555977 > 3¹⁶⁵ [i]