MinT - Best Known (17−7, 17, s)-Nets in Base 49

Best Known (17−7, 17, s)-Nets in Base 49

(17−7, 17, 2451)-Net over F₄₉ — Constructive and digital

Digital (10, 17, 2451)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 50)-net over F₄₉, using
  - s-reduction based on digital (0, 0, s)-net over F₄₉ with arbitrarily large s, using
    - arbitrary set of 49⁰ points [i]
2. digital (0, 0, 50)-net over F₄₉ (see above)
3. digital (0, 0, 50)-net over F₄₉ (see above)
4. digital (0, 0, 50)-net over F₄₉ (see above)
5. digital (0, 0, 50)-net over F₄₉ (see above)
6. digital (0, 0, 50)-net over F₄₉ (see above)
7. digital (0, 0, 50)-net over F₄₉ (see above)
8. digital (0, 0, 50)-net over F₄₉ (see above)
9. digital (0, 0, 50)-net over F₄₉ (see above)
10. digital (0, 0, 50)-net over F₄₉ (see above)
11. digital (0, 0, 50)-net over F₄₉ (see above)
12. digital (0, 0, 50)-net over F₄₉ (see above)
13. digital (0, 0, 50)-net over F₄₉ (see above)
14. digital (0, 0, 50)-net over F₄₉ (see above)
15. digital (0, 0, 50)-net over F₄₉ (see above)
16. digital (0, 0, 50)-net over F₄₉ (see above)
17. digital (0, 0, 50)-net over F₄₉ (see above)
18. digital (0, 0, 50)-net over F₄₉ (see above)
19. digital (0, 0, 50)-net over F₄₉ (see above)
20. digital (0, 0, 50)-net over F₄₉ (see above)
21. digital (0, 0, 50)-net over F₄₉ (see above)
22. digital (0, 0, 50)-net over F₄₉ (see above)
23. digital (0, 0, 50)-net over F₄₉ (see above)
24. digital (0, 0, 50)-net over F₄₉ (see above)
25. digital (0, 0, 50)-net over F₄₉ (see above)
26. digital (0, 0, 50)-net over F₄₉ (see above)
27. digital (0, 0, 50)-net over F₄₉ (see above)
28. digital (0, 0, 50)-net over F₄₉ (see above)
29. digital (0, 0, 50)-net over F₄₉ (see above)
30. digital (0, 0, 50)-net over F₄₉ (see above)
31. digital (0, 0, 50)-net over F₄₉ (see above)
32. digital (0, 0, 50)-net over F₄₉ (see above)
33. digital (0, 0, 50)-net over F₄₉ (see above)
34. digital (0, 0, 50)-net over F₄₉ (see above)
35. digital (0, 0, 50)-net over F₄₉ (see above)
36. digital (0, 0, 50)-net over F₄₉ (see above)
37. digital (0, 0, 50)-net over F₄₉ (see above)
38. digital (0, 0, 50)-net over F₄₉ (see above)
39. digital (0, 0, 50)-net over F₄₉ (see above)
40. digital (0, 0, 50)-net over F₄₉ (see above)
41. digital (0, 0, 50)-net over F₄₉ (see above)
42. digital (0, 0, 50)-net over F₄₉ (see above)
43. 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]
44. digital (0, 1, 50)-net over F₄₉ (see above)
45. digital (0, 1, 50)-net over F₄₉ (see above)
46. digital (0, 1, 50)-net over F₄₉ (see above)
47. digital (0, 2, 50)-net over F₄₉, using
  - kÂ =Â 2 construction [i]
48. 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]
49. digital (1, 8, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(17−7, 17, 4805)-Net over F₄₉ — Digital

Digital (10, 17, 4805)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁷, 4805, F₄₉, 7) (dual of [4805, 4788, 8]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(49⁴, 2402, F₄₉, 3) (dual of [2402, 2398, 4]-code or 2402-cap in PG(3,49)), using
    - ovoid in PG(3,Â 49) [i]
  2. linear OA(49¹³, 2403, F₄₉, 7) (dual of [2403, 2390, 8]-code), using
    - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
      1. linear OA(49¹³, 2401, F₄₉, 7) (dual of [2401, 2388, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OA(49¹¹, 2401, F₄₉, 6) (dual of [2401, 2390, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(17−7, 17, large)-Net in Base 49 — Upper bound on s

There is no (10, 17, large)-net in base 49, because

5 times m-reduction [i] would yield (10, 12, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (17−7, 17, s)-Nets in Base 49

(17−7, 17, 2451)-Net over F49 — Constructive and digital

(17−7, 17, 4805)-Net over F49 — Digital

(17−7, 17, large)-Net in Base 49 — Upper bound on s

(17−7, 17, 2451)-Net over F₄₉ — Constructive and digital

(17−7, 17, 4805)-Net over F₄₉ — Digital