MinT - Best Known (138, 138+45, s)-Nets in Base 3

Best Known (138, 138+45, s)-Nets in Base 3

(138, 138+45, 328)-Net over F₃ — Constructive and digital

Digital (138, 183, 328)-net over F₃, using

1 times m-reduction [i] based on digital (138, 184, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 46, 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]

(138, 138+45, 845)-Net over F₃ — Digital

Digital (138, 183, 845)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+45, 40057)-Net in Base 3 — Upper bound on s

There is no (138, 183, 40058)-net in base 3, because

1 times m-reduction [i] would yield (138, 182, 40058)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 685 776583 935409 346173 099001 006681 428882 048758 301870 070356 972197 051020 695081 725967 640109 > 3¹⁸² [i]