MinT - Best Known (16−5, 16, s)-Nets in Base 27

Best Known (16−5, 16, s)-Nets in Base 27

(16−5, 16, 20412)-Net over F₂₇ — Constructive and digital

Digital (11, 16, 20412)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 756)-net over F₂₇, using
  - s-reduction based on digital (0, 0, s)-net over F₂₇ with arbitrarily large s, using
    - arbitrary set of 27⁰ points [i]
2. digital (0, 0, 756)-net over F₂₇ (see above)
3. digital (0, 0, 756)-net over F₂₇ (see above)
4. digital (0, 0, 756)-net over F₂₇ (see above)
5. digital (0, 0, 756)-net over F₂₇ (see above)
6. digital (0, 0, 756)-net over F₂₇ (see above)
7. digital (0, 0, 756)-net over F₂₇ (see above)
8. digital (0, 0, 756)-net over F₂₇ (see above)
9. digital (0, 0, 756)-net over F₂₇ (see above)
10. digital (0, 0, 756)-net over F₂₇ (see above)
11. digital (0, 0, 756)-net over F₂₇ (see above)
12. digital (0, 0, 756)-net over F₂₇ (see above)
13. digital (0, 0, 756)-net over F₂₇ (see above)
14. digital (0, 0, 756)-net over F₂₇ (see above)
15. digital (0, 0, 756)-net over F₂₇ (see above)
16. digital (0, 0, 756)-net over F₂₇ (see above)
17. digital (0, 0, 756)-net over F₂₇ (see above)
18. digital (0, 0, 756)-net over F₂₇ (see above)
19. digital (0, 0, 756)-net over F₂₇ (see above)
20. digital (0, 0, 756)-net over F₂₇ (see above)
21. digital (0, 0, 756)-net over F₂₇ (see above)
22. digital (0, 0, 756)-net over F₂₇ (see above)
23. digital (0, 1, 756)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
24. digital (0, 1, 756)-net over F₂₇ (see above)
25. digital (0, 1, 756)-net over F₂₇ (see above)
26. digital (1, 3, 756)-net over F₂₇, using
  - s-reduction based on digital (1, 3, 757)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
27. digital (5, 10, 756)-net over F₂₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 28)-net over F₂₇, using
      - s-reduction based on digital (0, 0, s)-net over F₂₇ with arbitrarily large s (see above)
    2. digital (0, 0, 28)-net over F₂₇ (see above)
    3. digital (0, 0, 28)-net over F₂₇ (see above)
    4. digital (0, 0, 28)-net over F₂₇ (see above)
    5. digital (0, 0, 28)-net over F₂₇ (see above)
    6. digital (0, 0, 28)-net over F₂₇ (see above)
    7. digital (0, 0, 28)-net over F₂₇ (see above)
    8. digital (0, 0, 28)-net over F₂₇ (see above)
    9. digital (0, 0, 28)-net over F₂₇ (see above)
    10. digital (0, 0, 28)-net over F₂₇ (see above)
    11. digital (0, 0, 28)-net over F₂₇ (see above)
    12. digital (0, 0, 28)-net over F₂₇ (see above)
    13. digital (0, 0, 28)-net over F₂₇ (see above)
    14. digital (0, 0, 28)-net over F₂₇ (see above)
    15. digital (0, 0, 28)-net over F₂₇ (see above)
    16. digital (0, 0, 28)-net over F₂₇ (see above)
    17. digital (0, 0, 28)-net over F₂₇ (see above)
    18. digital (0, 0, 28)-net over F₂₇ (see above)
    19. digital (0, 0, 28)-net over F₂₇ (see above)
    20. digital (0, 0, 28)-net over F₂₇ (see above)
    21. digital (0, 0, 28)-net over F₂₇ (see above)
    22. digital (0, 0, 28)-net over F₂₇ (see above)
    23. digital (0, 1, 28)-net over F₂₇, using
      - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s (see above)
    24. digital (0, 1, 28)-net over F₂₇ (see above)
    25. digital (0, 1, 28)-net over F₂₇ (see above)
    26. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    27. digital (0, 5, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]

(16−5, 16, 45243)-Net over F₂₇ — Digital

Digital (11, 16, 45243)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(16−5, 16, large)-Net in Base 27 — Upper bound on s

There is no (11, 16, large)-net in base 27, because

3 times m-reduction [i] would yield (11, 13, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (16−5, 16, s)-Nets in Base 27

(16−5, 16, 20412)-Net over F27 — Constructive and digital

(16−5, 16, 45243)-Net over F27 — Digital

(16−5, 16, large)-Net in Base 27 — Upper bound on s

(16−5, 16, 20412)-Net over F₂₇ — Constructive and digital

(16−5, 16, 45243)-Net over F₂₇ — Digital