MinT - Best Known (40−27, 40, s)-Nets in Base 49

Best Known (40−27, 40, s)-Nets in Base 49

(40−27, 40, 100)-Net over F₄₉ — Constructive and digital

Digital (13, 40, 100)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 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, 27, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(40−27, 40, 13885)-Net in Base 49 — Upper bound on s

There is no (13, 40, 13886)-net in base 49, because

1 times m-reduction [i] would yield (13, 39, 13886)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 827766 134158 573939 142834 966602 188277 745732 249304 717191 080050 275617 > 49³⁹ [i]