MinT - Information on Result #1205552

Information on Result #1205552

Digital (80, 88, large)-net over F₉, using trace code for nets based on digital (36, 44, 8388602)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81⁴⁴, 8388602, F₈₁, 12, 8) (dual of [(8388602, 12), 100663180, 9]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(81⁴⁴, large, F₈₁, 4, 8), 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(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
    71. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
    72. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
    73. linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
    74. 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]
    75. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
    76. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
    77. linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
    78. 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]
    79. linear OOA(81⁴, 132861, F₈₁, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code) (see above)
    80. 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]
    81. linear OOA(81²², 132861, F₈₁, 4, 8) (dual of [(132861, 4), 531422, 9]-NRT-code), using
      - OOA 4-folding [i] based on linear OA(81²², 531444, F₈₁, 8) (dual of [531444, 531422, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(81²², 531441, F₈₁, 8) (dual of [531441, 531419, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(81¹⁹, 531441, F₈₁, 7) (dual of [531441, 531422, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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		Method
1	Digital (81, 88, large)-net over F₉	[i]	t-Expansion
2	Digital (82, 88, large)-net over F₉	[i]
3	Digital (83, 88, large)-net over F₉	[i]
4	Digital (84, 88, large)-net over F₉	[i]
5	Digital (81, 89, large)-net over F₉	[i]	Net Duplication
6	Digital (82, 90, large)-net over F₉	[i]
7	Digital (83, 91, large)-net over F₉	[i]
8	Digital (84, 92, large)-net over F₉	[i]
9	Digital (80, 87, large)-net over F₉	[i]	m-Reduction
10	Digital (80, 86, large)-net over F₉	[i]
11	Digital (80, 85, large)-net over F₉	[i]
12	Digital (80, 84, large)-net over F₉	[i]
13	Digital (80, 83, large)-net over F₉	[i]