MinT - Best Known (137, 137+45, s)-Nets in Base 3

Best Known (137, 137+45, s)-Nets in Base 3

(137, 137+45, 328)-Net over F₃ — Constructive and digital

Digital (137, 182, 328)-net over F₃, using

3² times duplication [i] based on 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]

(137, 137+45, 823)-Net over F₃ — Digital

Digital (137, 182, 823)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+45, 38105)-Net in Base 3 — Upper bound on s

There is no (137, 182, 38106)-net in base 3, because

1 times m-reduction [i] would yield (137, 181, 38106)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 228 631094 851234 718646 174288 019425 651426 679629 833513 390232 502273 747688 177900 351713 140333 > 3¹⁸¹ [i]