MinT - Best Known (138, 138+109, s)-Nets in Base 3

Best Known (138, 138+109, s)-Nets in Base 3

(138, 138+109, 128)-Net over F₃ — Constructive and digital

Digital (138, 247, 128)-net over F₃, using

3 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+109, 171)-Net over F₃ — Digital

Digital (138, 247, 171)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+109, 1510)-Net in Base 3 — Upper bound on s

There is no (138, 247, 1511)-net in base 3, because

1 times m-reduction [i] would yield (138, 246, 1511)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2384 228544 473412 590937 103001 420076 368820 815393 522616 148053 640932 873453 676309 671632 636289 347593 072508 626200 366086 454309 > 3²⁴⁶ [i]