MinT - Best Known (180−45, 180, s)-Nets in Base 3

Best Known (180−45, 180, s)-Nets in Base 3

(180−45, 180, 328)-Net over F₃ — Constructive and digital

Digital (135, 180, 328)-net over F₃, using

trace code for nets [i] based on digital (0, 45, 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]

(180−45, 180, 781)-Net over F₃ — Digital

Digital (135, 180, 781)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−45, 180, 34481)-Net in Base 3 — Upper bound on s

There is no (135, 180, 34482)-net in base 3, because

1 times m-reduction [i] would yield (135, 179, 34482)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 402771 289974 571778 747514 780636 903442 635384 002818 735265 648472 783758 209638 544117 598717 > 3¹⁷⁹ [i]