MinT - Best Known (51−19, 51, s)-Nets in Base 49

Best Known (51−19, 51, s)-Nets in Base 49

(51−19, 51, 500)-Net over F₄₉ — Constructive and digital

Digital (32, 51, 500)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 50)-net over F₄₉, using
  - s-reduction based on digital (0, 1, s)-net over F₄₉ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 2, 50)-net over F₄₉, using
  - kÂ =Â 2 construction [i]
3. digital (0, 2, 50)-net over F₄₉ (see above)
4. digital (0, 2, 50)-net over F₄₉ (see above)
5. digital (0, 3, 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]
6. digital (0, 3, 50)-net over F₄₉ (see above)
7. digital (0, 4, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
8. digital (0, 6, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
9. digital (0, 9, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
10. digital (0, 19, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(51−19, 51, 9687)-Net over F₄₉ — Digital

Digital (32, 51, 9687)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−19, 51, large)-Net in Base 49 — Upper bound on s

There is no (32, 51, large)-net in base 49, because

17 times m-reduction [i] would yield (32, 34, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]