MinT - Information on Result #2705526

Information on Result #2705526

Linear OOA(81⁴¹, large, F₈₁, 3, 5), using 7 times NRT-code embedding in larger space based on linear OOA(81²⁰, large, F₈₁, 3, 5), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 3, 0) for arbitrarily large s, using
    - OOA with only one run / dual of NRT-code without redundancy [i]
2. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
3. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
4. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
5. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
6. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
7. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
8. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
9. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
10. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
11. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
12. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
13. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
14. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
15. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
16. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
17. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
18. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
19. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
20. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
21. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
22. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
23. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
24. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
25. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
26. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
27. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
28. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
29. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
30. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
31. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
32. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
33. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
34. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
35. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
36. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
37. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
38. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
39. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
40. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
41. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
42. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
43. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
44. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
45. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
46. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
47. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
48. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
49. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
50. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
51. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
52. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
53. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
54. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
55. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
56. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
57. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
58. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
59. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
60. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
61. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
62. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
63. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
64. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
65. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
66. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
67. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
68. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
69. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
70. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
71. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
72. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
73. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
74. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
75. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
76. linear OOA(81⁰, 265721, F₈₁, 3, 0) (dual of [(265721, 3), 797163, 1]-NRT-code) (see above)
77. linear OOA(81¹, 265721, F₈₁, 3, 1) (dual of [(265721, 3), 797162, 2]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 3, 1) for arbitrarily large s, using
    - appending 2 arbitrary columns [i] based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
      - parity-check code [i]
78. linear OOA(81¹, 265721, F₈₁, 3, 1) (dual of [(265721, 3), 797162, 2]-NRT-code) (see above)
79. linear OOA(81¹, 265721, F₈₁, 3, 1) (dual of [(265721, 3), 797162, 2]-NRT-code) (see above)
80. linear OOA(81⁴, 265721, F₈₁, 3, 2) (dual of [(265721, 3), 797159, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81⁴, 538084, F₈₁, 3, 2) (dual of [(538084, 3), 1614248, 3]-NRT-code), using
    - appending 1 arbitrary column [i] based on linear OOA(81⁴, 538084, F₈₁, 2, 2) (dual of [(538084, 2), 1076164, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(81⁴, 538084, F₈₁, 2) (dual of [538084, 538080, 3]-code), using
        Hamming code H(4,81) [i]
81. linear OOA(81¹³, 265721, F₈₁, 3, 5) (dual of [(265721, 3), 797150, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(81¹³, 531443, F₈₁, 5) (dual of [531443, 531430, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(81¹³, 531444, F₈₁, 5) (dual of [531444, 531431, 6]-code), using
      - constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 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(81⁴², large, F₈₁, 3, 5)	[i]		OOA Duplication
2	Linear OOA(81⁴³, large, F₈₁, 3, 5)	[i]		OOA Duplication