MinT - Best Known (38−21, 38, s)-Nets in Base 49

Best Known (38−21, 38, s)-Nets in Base 49

(38−21, 38, 150)-Net over F₄₉ — Constructive and digital

Digital (17, 38, 150)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 7, 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, 10, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
3. digital (0, 21, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(38−21, 38, 292)-Net over F₄₉ — Digital

Digital (17, 38, 292)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(38−21, 38, 169218)-Net in Base 49 — Upper bound on s

There is no (17, 38, 169219)-net in base 49, because

1 times m-reduction [i] would yield (17, 37, 169219)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 344 557270 736318 267237 946831 078397 432181 498277 555557 704459 576225 > 49³⁷ [i]