MinT - Best Known (10, 10+5, s)-Nets in Base 27

Best Known (10, 10+5, s)-Nets in Base 27

(10, 10+5, 10233)-Net over F₂₇ — Constructive and digital

Digital (10, 15, 10233)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 379)-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, 379)-net over F₂₇ (see above)
3. digital (0, 0, 379)-net over F₂₇ (see above)
4. digital (0, 0, 379)-net over F₂₇ (see above)
5. digital (0, 0, 379)-net over F₂₇ (see above)
6. digital (0, 0, 379)-net over F₂₇ (see above)
7. digital (0, 0, 379)-net over F₂₇ (see above)
8. digital (0, 0, 379)-net over F₂₇ (see above)
9. digital (0, 0, 379)-net over F₂₇ (see above)
10. digital (0, 0, 379)-net over F₂₇ (see above)
11. digital (0, 0, 379)-net over F₂₇ (see above)
12. digital (0, 0, 379)-net over F₂₇ (see above)
13. digital (0, 0, 379)-net over F₂₇ (see above)
14. digital (0, 0, 379)-net over F₂₇ (see above)
15. digital (0, 0, 379)-net over F₂₇ (see above)
16. digital (0, 0, 379)-net over F₂₇ (see above)
17. digital (0, 0, 379)-net over F₂₇ (see above)
18. digital (0, 0, 379)-net over F₂₇ (see above)
19. digital (0, 0, 379)-net over F₂₇ (see above)
20. digital (0, 0, 379)-net over F₂₇ (see above)
21. digital (0, 0, 379)-net over F₂₇ (see above)
22. digital (0, 0, 379)-net over F₂₇ (see above)
23. digital (0, 1, 379)-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, 379)-net over F₂₇ (see above)
25. digital (0, 1, 379)-net over F₂₇ (see above)
26. digital (1, 3, 379)-net over F₂₇, using
  - s-reduction based on digital (1, 3, 757)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
27. digital (4, 9, 379)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 351)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]

(10, 10+5, 19849)-Net over F₂₇ — Digital

Digital (10, 15, 19849)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(10, 10+5, large)-Net in Base 27 — Upper bound on s

There is no (10, 15, large)-net in base 27, because

3 times m-reduction [i] would yield (10, 12, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (10, 10+5, s)-Nets in Base 27

(10, 10+5, 10233)-Net over F27 — Constructive and digital

(10, 10+5, 19849)-Net over F27 — Digital

(10, 10+5, large)-Net in Base 27 — Upper bound on s

(10, 10+5, 10233)-Net over F₂₇ — Constructive and digital

(10, 10+5, 19849)-Net over F₂₇ — Digital