MinT - Best Known (5, 5+5, s)-Nets in Base 49

Best Known (5, 5+5, s)-Nets in Base 49

(5, 5+5, 2450)-Net over F₄₉ — Constructive and digital

Digital (5, 10, 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, 0, 50)-net over F₄₉ (see above)
44. digital (0, 0, 50)-net over F₄₉ (see above)
45. 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]
46. digital (0, 1, 50)-net over F₄₉ (see above)
47. digital (0, 1, 50)-net over F₄₉ (see above)
48. digital (0, 2, 50)-net over F₄₉, using
  - kÂ =Â 2 construction [i]
49. digital (0, 5, 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]

(5, 5+5, 4453)-Net over F₄₉ — Digital

Digital (5, 10, 4453)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁰, 4453, F₄₉, 5) (dual of [4453, 4443, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49¹⁰, 4706, F₄₉, 5) (dual of [4706, 4696, 6]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(49³, 2353, F₄₉, 2) (dual of [2353, 2350, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(49³, 2451, F₄₉, 2) (dual of [2451, 2448, 3]-code), using
        Hamming code H(3,49) [i]
    2. linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
      - a code by Danev and Olsson [i]

(5, 5+5, 1188929)-Net in Base 49 — Upper bound on s

There is no (5, 10, 1188930)-net in base 49, because

1 times m-reduction [i] would yield (5, 9, 1188930)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1628 416319 509441 > 49⁹ [i]

Best Known (5, 5+5, s)-Nets in Base 49

(5, 5+5, 2450)-Net over F49 — Constructive and digital

(5, 5+5, 4453)-Net over F49 — Digital

(5, 5+5, 1188929)-Net in Base 49 — Upper bound on s

(5, 5+5, 2450)-Net over F₄₉ — Constructive and digital

(5, 5+5, 4453)-Net over F₄₉ — Digital