MinT - Information on Result #3178265

Information on Result #3178265

Digital (59, 64, large)-net over F₆₄, using 5 times m-reduction based on digital (59, 69, large)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 131073)-net over F₆₄, using
  - s-reduction based on digital (0, 0, s)-net over F₆₄ for arbitrarily large s, using
    - arbitrary set of 64⁰ points [i]
2. digital (0, 0, 131073)-net over F₆₄ (see above)
3. digital (0, 0, 131073)-net over F₆₄ (see above)
4. digital (0, 0, 131073)-net over F₆₄ (see above)
5. digital (0, 0, 131073)-net over F₆₄ (see above)
6. digital (0, 0, 131073)-net over F₆₄ (see above)
7. digital (0, 0, 131073)-net over F₆₄ (see above)
8. digital (0, 0, 131073)-net over F₆₄ (see above)
9. digital (0, 0, 131073)-net over F₆₄ (see above)
10. digital (0, 0, 131073)-net over F₆₄ (see above)
11. digital (0, 0, 131073)-net over F₆₄ (see above)
12. digital (0, 0, 131073)-net over F₆₄ (see above)
13. digital (0, 0, 131073)-net over F₆₄ (see above)
14. digital (0, 0, 131073)-net over F₆₄ (see above)
15. digital (0, 0, 131073)-net over F₆₄ (see above)
16. digital (0, 0, 131073)-net over F₆₄ (see above)
17. digital (0, 0, 131073)-net over F₆₄ (see above)
18. digital (0, 0, 131073)-net over F₆₄ (see above)
19. digital (0, 0, 131073)-net over F₆₄ (see above)
20. digital (0, 0, 131073)-net over F₆₄ (see above)
21. digital (0, 0, 131073)-net over F₆₄ (see above)
22. digital (0, 0, 131073)-net over F₆₄ (see above)
23. digital (0, 0, 131073)-net over F₆₄ (see above)
24. digital (0, 0, 131073)-net over F₆₄ (see above)
25. digital (0, 0, 131073)-net over F₆₄ (see above)
26. digital (0, 0, 131073)-net over F₆₄ (see above)
27. digital (0, 0, 131073)-net over F₆₄ (see above)
28. digital (0, 0, 131073)-net over F₆₄ (see above)
29. digital (0, 0, 131073)-net over F₆₄ (see above)
30. digital (0, 0, 131073)-net over F₆₄ (see above)
31. digital (0, 0, 131073)-net over F₆₄ (see above)
32. digital (0, 0, 131073)-net over F₆₄ (see above)
33. digital (0, 0, 131073)-net over F₆₄ (see above)
34. digital (0, 0, 131073)-net over F₆₄ (see above)
35. digital (0, 0, 131073)-net over F₆₄ (see above)
36. digital (0, 0, 131073)-net over F₆₄ (see above)
37. digital (0, 0, 131073)-net over F₆₄ (see above)
38. digital (0, 0, 131073)-net over F₆₄ (see above)
39. digital (0, 0, 131073)-net over F₆₄ (see above)
40. digital (0, 0, 131073)-net over F₆₄ (see above)
41. digital (0, 0, 131073)-net over F₆₄ (see above)
42. digital (0, 0, 131073)-net over F₆₄ (see above)
43. digital (0, 0, 131073)-net over F₆₄ (see above)
44. digital (0, 0, 131073)-net over F₆₄ (see above)
45. digital (0, 0, 131073)-net over F₆₄ (see above)
46. digital (0, 0, 131073)-net over F₆₄ (see above)
47. digital (0, 0, 131073)-net over F₆₄ (see above)
48. digital (0, 0, 131073)-net over F₆₄ (see above)
49. digital (0, 0, 131073)-net over F₆₄ (see above)
50. digital (0, 0, 131073)-net over F₆₄ (see above)
51. digital (0, 0, 131073)-net over F₆₄ (see above)
52. digital (0, 0, 131073)-net over F₆₄ (see above)
53. digital (0, 0, 131073)-net over F₆₄ (see above)
54. digital (0, 0, 131073)-net over F₆₄ (see above)
55. digital (0, 1, 131073)-net over F₆₄, using
  - s-reduction based on digital (0, 1, s)-net over F₆₄ for arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
56. digital (0, 1, 131073)-net over F₆₄ (see above)
57. digital (0, 1, 131073)-net over F₆₄ (see above)
58. digital (0, 1, 131073)-net over F₆₄ (see above)
59. digital (0, 1, 131073)-net over F₆₄ (see above)
60. digital (2, 4, 131073)-net over F₆₄, using
  - s-reduction based on digital (2, 4, 266305)-net over F₆₄, using
    - kÂ =Â 2 construction [i]
61. digital (2, 4, 131073)-net over F₆₄ (see above)
62. digital (3, 6, 131073)-net over F₆₄, using
  - s-reduction based on digital (3, 6, 270402)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶, 270402, F₆₄, 3, 3) (dual of [(270402, 3), 811200, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(64⁶, 270402, F₆₄, 2, 3) (dual of [(270402, 2), 540798, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁶, 270402, F₆₄, 3) (dual of [270402, 270396, 4]-code or 270402-cap in PG(5,64)), using
        large caps in PG(5,Â b) [i]
63. digital (8, 13, 131073)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64¹³, 131073, F₆₄, 5, 5) (dual of [(131073, 5), 655352, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
      - constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
64. digital (27, 37, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Digital (60, 64, large)-net over F₆₄	[i]		t-Expansion
2	Digital (61, 64, large)-net over F₆₄	[i]		t-Expansion