MinT - Best Known (138−37, 138, s)-Nets in Base 5

Best Known (138−37, 138, s)-Nets in Base 5

(138−37, 138, 408)-Net over F₅ — Constructive and digital

Digital (101, 138, 408)-net over F₅, using

4 times m-reduction [i] based on digital (101, 142, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 71, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(138−37, 138, 1724)-Net over F₅ — Digital

Digital (101, 138, 1724)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−37, 138, 394451)-Net in Base 5 — Upper bound on s

There is no (101, 138, 394452)-net in base 5, because

1 times m-reduction [i] would yield (101, 137, 394452)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 573972 883776 495053 076469 563828 157179 612084 833711 055986 863400 707226 318547 257124 454210 777483 621825 > 5¹³⁷ [i]