MinT - Best Known (138, 138+107, s)-Nets in Base 3

Best Known (138, 138+107, s)-Nets in Base 3

(138, 138+107, 128)-Net over F₃ — Constructive and digital

Digital (138, 245, 128)-net over F₃, using

5 times m-reduction [i] based on digital (138, 250, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 125, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(138, 138+107, 175)-Net over F₃ — Digital

Digital (138, 245, 175)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+107, 1567)-Net in Base 3 — Upper bound on s

There is no (138, 245, 1568)-net in base 3, because

1 times m-reduction [i] would yield (138, 244, 1568)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 266 169230 934931 822033 645838 052300 712917 828509 270708 401247 965175 368531 156713 164323 556220 625867 100545 577346 608632 809537 > 3²⁴⁴ [i]