MinT - Best Known (20, 20+25, s)-Nets in Base 49

Best Known (20, 20+25, s)-Nets in Base 49

(20, 20+25, 150)-Net over F₄₉ — Constructive and digital

Digital (20, 45, 150)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 8, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
      - the rational function field F₄₉(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 12, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
3. digital (0, 25, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(20, 20+25, 314)-Net over F₄₉ — Digital

Digital (20, 45, 314)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(20, 20+25, 173576)-Net in Base 49 — Upper bound on s

There is no (20, 45, 173577)-net in base 49, because

1 times m-reduction [i] would yield (20, 44, 173577)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 233 688131 642629 669828 990626 613427 425319 139422 150541 918014 619200 193706 933825 > 49⁴⁴ [i]