MinT - Best Known (10, 10+5, s)-Nets in Base 49

Best Known (10, 10+5, s)-Nets in Base 49

(10, 10+5, 60074)-Net over F₄₉ — Constructive and digital

Digital (10, 15, 60074)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 1226)-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, 1226)-net over F₄₉ (see above)
3. digital (0, 0, 1226)-net over F₄₉ (see above)
4. digital (0, 0, 1226)-net over F₄₉ (see above)
5. digital (0, 0, 1226)-net over F₄₉ (see above)
6. digital (0, 0, 1226)-net over F₄₉ (see above)
7. digital (0, 0, 1226)-net over F₄₉ (see above)
8. digital (0, 0, 1226)-net over F₄₉ (see above)
9. digital (0, 0, 1226)-net over F₄₉ (see above)
10. digital (0, 0, 1226)-net over F₄₉ (see above)
11. digital (0, 0, 1226)-net over F₄₉ (see above)
12. digital (0, 0, 1226)-net over F₄₉ (see above)
13. digital (0, 0, 1226)-net over F₄₉ (see above)
14. digital (0, 0, 1226)-net over F₄₉ (see above)
15. digital (0, 0, 1226)-net over F₄₉ (see above)
16. digital (0, 0, 1226)-net over F₄₉ (see above)
17. digital (0, 0, 1226)-net over F₄₉ (see above)
18. digital (0, 0, 1226)-net over F₄₉ (see above)
19. digital (0, 0, 1226)-net over F₄₉ (see above)
20. digital (0, 0, 1226)-net over F₄₉ (see above)
21. digital (0, 0, 1226)-net over F₄₉ (see above)
22. digital (0, 0, 1226)-net over F₄₉ (see above)
23. digital (0, 0, 1226)-net over F₄₉ (see above)
24. digital (0, 0, 1226)-net over F₄₉ (see above)
25. digital (0, 0, 1226)-net over F₄₉ (see above)
26. digital (0, 0, 1226)-net over F₄₉ (see above)
27. digital (0, 0, 1226)-net over F₄₉ (see above)
28. digital (0, 0, 1226)-net over F₄₉ (see above)
29. digital (0, 0, 1226)-net over F₄₉ (see above)
30. digital (0, 0, 1226)-net over F₄₉ (see above)
31. digital (0, 0, 1226)-net over F₄₉ (see above)
32. digital (0, 0, 1226)-net over F₄₉ (see above)
33. digital (0, 0, 1226)-net over F₄₉ (see above)
34. digital (0, 0, 1226)-net over F₄₉ (see above)
35. digital (0, 0, 1226)-net over F₄₉ (see above)
36. digital (0, 0, 1226)-net over F₄₉ (see above)
37. digital (0, 0, 1226)-net over F₄₉ (see above)
38. digital (0, 0, 1226)-net over F₄₉ (see above)
39. digital (0, 0, 1226)-net over F₄₉ (see above)
40. digital (0, 0, 1226)-net over F₄₉ (see above)
41. digital (0, 0, 1226)-net over F₄₉ (see above)
42. digital (0, 0, 1226)-net over F₄₉ (see above)
43. digital (0, 0, 1226)-net over F₄₉ (see above)
44. digital (0, 0, 1226)-net over F₄₉ (see above)
45. digital (0, 1, 1226)-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, 1226)-net over F₄₉ (see above)
47. digital (0, 1, 1226)-net over F₄₉ (see above)
48. digital (1, 3, 1226)-net over F₄₉, using
  - s-reduction based on digital (1, 3, 2451)-net over F₄₉, using
    - kÂ =Â 2 construction [i]
49. digital (4, 9, 1226)-net over F₄₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 50)-net over F₄₉, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 1176)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49⁷, 1176, F₄₉, 5, 5) (dual of [(1176, 5), 5873, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(10, 10+5, 117747)-Net over F₄₉ — Digital

Digital (10, 15, 117747)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁵, 117747, F₄₉, 5) (dual of [117747, 117732, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 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]
  2. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  3. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  4. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  5. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  6. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  7. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  8. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  9. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  10. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  11. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  12. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  13. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  14. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  15. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  16. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  17. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  18. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  19. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  20. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  21. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  22. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  23. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  24. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  25. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  26. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  27. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  28. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  29. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  30. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  31. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  32. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  33. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  34. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  35. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  36. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  37. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  38. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  39. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  40. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  41. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  42. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  43. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  44. linear OA(49⁰, 2403, F₄₉, 0) (dual of [2403, 2403, 1]-code) (see above)
  45. linear OA(49¹, 2403, F₄₉, 1) (dual of [2403, 2402, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  46. linear OA(49¹, 2403, F₄₉, 1) (dual of [2403, 2402, 2]-code) (see above)
  47. linear OA(49¹, 2403, F₄₉, 1) (dual of [2403, 2402, 2]-code) (see above)
  48. linear OA(49³, 2403, F₄₉, 2) (dual of [2403, 2400, 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]
  49. linear OA(49⁹, 2403, F₄₉, 5) (dual of [2403, 2394, 6]-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OA(49², 50, F₄₉, 2) (dual of [50, 48, 3]-code or 50-arc in PG(1,49)), using
        extended Reedâ€“Solomon code RS_e(48,49) [i]
        Hamming code H(2,49) [i]
        algebraic-geometric code AG(F, Q+22P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using the rational function field F₄₉(x) [i]
      2. linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(10, 10+5, large)-Net in Base 49 — Upper bound on s

There is no (10, 15, large)-net in base 49, because

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

Best Known (10, 10+5, s)-Nets in Base 49

(10, 10+5, 60074)-Net over F49 — Constructive and digital

(10, 10+5, 117747)-Net over F49 — Digital

(10, 10+5, large)-Net in Base 49 — Upper bound on s

(10, 10+5, 60074)-Net over F₄₉ — Constructive and digital

(10, 10+5, 117747)-Net over F₄₉ — Digital