MinT - Information on Result #2520716

Information on Result #2520716

Linear OOA(64⁵¹, large, F₆₄, 2, 7), using 9 times NRT-code embedding in larger space based on linear OOA(64³³, large, F₆₄, 2, 7), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(64⁰, s, F₆₄, 2, 0) for arbitrarily large s, using
    - OOA with only one run / dual of NRT-code without redundancy [i]
2. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
3. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
4. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
5. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
6. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
7. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
8. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
9. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
10. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
11. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
12. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
13. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
14. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
15. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
16. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
17. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
18. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
19. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
20. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
21. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
22. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
23. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
24. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
25. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
26. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
27. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
28. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
29. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
30. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
31. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
32. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
33. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
34. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
35. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
36. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
37. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
38. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
39. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
40. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
41. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
42. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
43. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
44. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
45. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
46. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
47. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
48. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
49. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
50. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
51. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
52. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
53. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
54. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
55. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
56. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
57. linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
58. linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(64¹, s, F₆₄, 2, 1) for arbitrarily large s, using
    - appending 1 arbitrary column [i] based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
      - parity-check code [i]
59. linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code) (see above)
60. linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code) (see above)
61. linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code) (see above)
62. linear OOA(64⁴, 131073, F₆₄, 2, 2) (dual of [(131073, 2), 262142, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(64⁴, 266305, F₆₄, 2, 2) (dual of [(266305, 2), 532606, 3]-NRT-code), using
    - appending kth column [i] based on linear OA(64⁴, 266305, F₆₄, 2) (dual of [266305, 266301, 3]-code), using
      - Hamming code H(4,64) [i]
63. linear OOA(64⁶, 131073, F₆₄, 2, 3) (dual of [(131073, 2), 262140, 4]-NRT-code), using
  - discarding factors / shortening the dual code 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]
64. linear OOA(64¹⁹, 131073, F₆₄, 2, 7) (dual of [(131073, 2), 262127, 8]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64¹⁹, 262146, F₆₄, 7) (dual of [262146, 262127, 8]-code), using
    - discarding factors / shortening the dual code based on linear OA(64¹⁹, 262147, F₆₄, 7) (dual of [262147, 262128, 8]-code), using
      - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(64¹⁹, 262144, F₆₄, 7) (dual of [262144, 262125, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

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	Linear OOA(64⁵², large, F₆₄, 2, 7)	[i]		OOA Duplication