MinT - Best Known (16−4, 16, s)-Nets in Base 81

Best Known (16−4, 16, s)-Nets in Base 81

(16−4, 16, large)-Net over F₈₁ — Constructive and digital

Digital (12, 16, large)-net over F₈₁, using

net defined by OOA [i] 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) with 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) with arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with 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) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(16−4, 16, large)-Net in Base 81 — Upper bound on s

There is no (12, 16, large)-net in base 81, because

2 times m-reduction [i] would yield (12, 14, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (16−4, 16, s)-Nets in Base 81

(16−4, 16, large)-Net over F81 — Constructive and digital

(16−4, 16, large)-Net in Base 81 — Upper bound on s

(16−4, 16, large)-Net over F₈₁ — Constructive and digital