MinT - Best Known (40−15, 40, s)-Nets in Base 49

Best Known (40−15, 40, s)-Nets in Base 49

(40−15, 40, 500)-Net over F₄₉ — Constructive and digital

Digital (25, 40, 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, 1, 50)-net over F₄₉ (see above)
3. digital (0, 1, 50)-net over F₄₉ (see above)
4. digital (0, 2, 50)-net over F₄₉, using
  - kÂ =Â 2 construction [i]
5. digital (0, 2, 50)-net over F₄₉ (see above)
6. 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]
7. digital (0, 3, 50)-net over F₄₉ (see above)
8. digital (0, 5, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
9. digital (0, 7, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
10. digital (0, 15, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(40−15, 40, 8506)-Net over F₄₉ — Digital

Digital (25, 40, 8506)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−15, 40, large)-Net in Base 49 — Upper bound on s

There is no (25, 40, large)-net in base 49, because

13 times m-reduction [i] would yield (25, 27, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]