MinT - Best Known (52−35, 52, s)-Nets in Base 49

Best Known (52−35, 52, s)-Nets in Base 49

(52−35, 52, 100)-Net over F₄₉ — Constructive and digital

Digital (17, 52, 100)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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, 35, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(52−35, 52, 103)-Net over F₄₉ — Digital

Digital (17, 52, 103)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−35, 52, 17582)-Net in Base 49 — Upper bound on s

There is no (17, 52, 17583)-net in base 49, because

1 times m-reduction [i] would yield (17, 51, 17583)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 158 533042 606285 465123 768742 926558 821924 297900 044477 767589 805836 738004 080333 783692 094929 > 49⁵¹ [i]