MinT - Best Known (16−7, 16, s)-Nets in Base 49

Best Known (16−7, 16, s)-Nets in Base 49

(16−7, 16, 2450)-Net over F₄₉ — Constructive and digital

Digital (9, 16, 2450)-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 (0, 7, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(16−7, 16, 2714)-Net over F₄₉ — Digital

Digital (9, 16, 2714)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁶, 2714, F₄₉, 7) (dual of [2714, 2698, 8]-code), using
- 308 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 29 times 0, 1, 277 times 0) [i] based on 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]

(16−7, 16, large)-Net in Base 49 — Upper bound on s

There is no (9, 16, large)-net in base 49, because

5 times m-reduction [i] would yield (9, 11, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (16−7, 16, s)-Nets in Base 49

(16−7, 16, 2450)-Net over F49 — Constructive and digital

(16−7, 16, 2714)-Net over F49 — Digital

(16−7, 16, large)-Net in Base 49 — Upper bound on s

(16−7, 16, 2450)-Net over F₄₉ — Constructive and digital

(16−7, 16, 2714)-Net over F₄₉ — Digital