MinT - Information on Result #2778498

Information on Result #2778498

Linear OOA(81⁸⁰, large, F₈₁, 4, 4), using strength reduction based on linear OOA(81⁸⁰, large, F₈₁, 4, 12), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 4, 0) for arbitrarily large s, using
    - OOA with only one run / dual of NRT-code without redundancy [i]
2. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
3. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
4. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
5. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
6. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
7. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
8. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
9. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
10. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
11. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
12. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
13. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
14. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
15. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
16. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
17. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
18. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
19. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
20. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
21. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
22. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
23. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
24. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
25. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
26. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
27. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
28. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
29. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
30. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
31. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
32. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
33. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
34. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
35. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
36. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
37. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
38. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
39. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
40. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
41. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
42. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
43. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
44. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
45. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
46. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
47. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
48. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
49. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
50. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
51. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
52. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
53. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
54. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
55. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
56. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
57. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
58. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
59. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
60. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
61. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
62. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
63. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
64. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
65. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
66. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
67. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
68. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
69. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
70. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 4, 1) for arbitrarily large s, using
    - appending 3 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]
71. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
72. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
73. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
74. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
75. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
76. linear OOA(81⁴, 132861, F₈₁, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81⁴, 538084, F₈₁, 4, 2) (dual of [(538084, 4), 2152332, 3]-NRT-code), using
    - appending 2 arbitrary columns [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]
77. linear OOA(81⁴, 132861, F₈₁, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code) (see above)
78. linear OOA(81⁶, 132861, F₈₁, 4, 3) (dual of [(132861, 4), 531438, 4]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81⁶, 538248, F₈₁, 4, 3) (dual of [(538248, 4), 2152986, 4]-NRT-code), using
    - appending 1 arbitrary column [i] based on linear OOA(81⁶, 538248, F₈₁, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(81⁶, 538248, F₈₁, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
79. linear OOA(81¹⁰, 132861, F₈₁, 4, 4) (dual of [(132861, 4), 531434, 5]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81¹⁰, 265722, F₈₁, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
      - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]
80. linear OOA(81¹⁶, 132861, F₈₁, 4, 6) (dual of [(132861, 4), 531428, 7]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(81¹⁶, 177148, F₈₁, 4, 6) (dual of [(177148, 4), 708576, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
      - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)
81. linear OOA(81³⁴, 132861, F₈₁, 4, 12) (dual of [(132861, 4), 531410, 13]-NRT-code), using
  - OOA 4-folding [i] based on linear OA(81³⁴, 531444, F₈₁, 12) (dual of [531444, 531410, 13]-code), using
    - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
      1. linear OA(81³⁴, 531441, F₈₁, 12) (dual of [531441, 531407, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      2. linear OA(81³¹, 531441, F₈₁, 11) (dual of [531441, 531410, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)

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(81⁸¹, large, F₈₁, 4, 4)	[i]		OOA Duplication
2	Linear OOA(81⁸², large, F₈₁, 4, 4)	[i]		OOA Duplication