MinT - Best Known (123−55, 123, s)-Nets in Base 5

Best Known (123−55, 123, s)-Nets in Base 5

(123−55, 123, 132)-Net over F₅ — Constructive and digital

Digital (68, 123, 132)-net over F₅, using

5 times m-reduction [i] based on digital (68, 128, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 64, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(123−55, 123, 181)-Net over F₅ — Digital

Digital (68, 123, 181)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−55, 123, 3912)-Net in Base 5 — Upper bound on s

There is no (68, 123, 3913)-net in base 5, because

1 times m-reduction [i] would yield (68, 122, 3913)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 18 850907 067934 012097 441364 951676 658012 950796 135249 827875 259708 590031 749381 143153 033789 > 5¹²² [i]