MinT - Information on Result #1179700

Information on Result #1179700

Digital (15, 20, large)-net over F₈₁, using net defined by OOA based on linear OOA(81²⁰, large, F₈₁, 5, 5), using

appending kth column [i] based on linear OOA(81²⁰, large, F₈₁, 4, 5), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 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, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  3. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  4. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  5. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  6. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  7. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  8. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  9. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  10. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  11. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  12. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  13. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  14. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  15. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  16. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  17. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  18. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  19. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  20. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  21. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  22. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  23. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  24. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  25. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  26. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  27. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  28. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  29. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  30. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  31. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  32. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  33. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  34. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  35. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  36. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  37. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  38. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  39. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  40. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  41. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  42. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  43. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  44. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  45. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  46. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  47. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  48. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  49. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  50. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  51. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  52. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  53. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  54. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  55. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  56. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  57. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  58. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  59. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  60. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  61. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  62. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  63. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  64. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  65. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  66. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  67. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  68. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  69. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  70. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  71. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  72. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  73. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  74. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  75. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  76. linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
  77. linear OOA(81¹, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 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]
  78. linear OOA(81¹, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 2]-NRT-code) (see above)
  79. linear OOA(81¹, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 2]-NRT-code) (see above)
  80. linear OOA(81⁴, 265721, F₈₁, 4, 2) (dual of [(265721, 4), 1062880, 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]
  81. linear OOA(81¹³, 265721, F₈₁, 4, 5) (dual of [(265721, 4), 1062871, 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		Method
1	Digital (16, 20, large)-net over F₈₁	[i]	t-Expansion
2	Digital (17, 20, large)-net over F₈₁	[i]	t-Expansion
3	Digital (16, 21, large)-net over F₈₁	[i]	Net Duplication
4	Digital (17, 22, large)-net over F₈₁	[i]
5	Digital (18, 23, large)-net over F₈₁	[i]
6	Digital (19, 24, large)-net over F₈₁	[i]
7	Digital (20, 25, large)-net over F₈₁	[i]
8	Digital (21, 26, large)-net over F₈₁	[i]
9	Digital (22, 27, large)-net over F₈₁	[i]
10	Digital (15, 19, large)-net over F₈₁	[i]	m-Reduction
11	Digital (15, 18, large)-net over F₈₁	[i]	m-Reduction