MinT - Information on Result #1179072

Information on Result #1179072

Digital (12, 16, large)-net over F₈₁, using net defined by OOA based on linear OOA(81¹⁶, large, F₈₁, 4, 4), using

appending kth column [i] based on linear OOA(81¹⁶, large, F₈₁, 3, 4), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 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, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  3. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  4. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  5. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  6. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  7. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  8. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  9. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  10. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  11. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  12. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  13. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  14. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  15. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  16. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  17. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  18. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  19. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  20. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  21. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  22. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  23. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  24. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  25. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  26. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  27. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  28. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  29. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  30. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  31. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  32. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  33. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  34. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  35. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  36. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  37. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  38. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  39. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  40. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  41. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  42. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  43. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  44. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  45. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  46. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  47. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  48. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  49. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  50. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  51. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  52. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  53. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  54. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  55. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  56. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  57. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  58. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  59. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  60. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  61. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  62. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  63. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  64. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  65. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  66. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  67. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  68. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  69. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  70. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  71. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  72. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  73. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  74. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  75. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  76. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  77. linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
  78. linear OOA(81¹, 265722, F₈₁, 3, 1) (dual of [(265722, 3), 797165, 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]
  79. linear OOA(81¹, 265722, F₈₁, 3, 1) (dual of [(265722, 3), 797165, 2]-NRT-code) (see above)
  80. linear OOA(81⁴, 265722, F₈₁, 3, 2) (dual of [(265722, 3), 797162, 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(81¹⁰, 265722, F₈₁, 3, 4) (dual of [(265722, 3), 797156, 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]

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 (13, 16, large)-net over F₈₁	[i]	t-Expansion
2	Digital (13, 17, large)-net over F₈₁	[i]	Net Duplication
3	Digital (14, 18, large)-net over F₈₁	[i]	Net Duplication
4	Digital (12, 15, large)-net over F₈₁	[i]	m-Reduction