MinT - Best Known (18, 23, s)-Nets in Base 81

Best Known (18, 23, s)-Nets in Base 81

(18, 23, large)-Net over F₈₁ — Constructive and digital

Digital (18, 23, large)-net over F₈₁, using

81³ times duplication [i] based on digital (15, 20, large)-net over F₈₁, using
- net defined by OOA [i] 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) with 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) with arbitrarily large s, using
        appending 3 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]
      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) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(18, 23, large)-Net in Base 81 — Upper bound on s

There is no (18, 23, large)-net in base 81, because

3 times m-reduction [i] would yield (18, 20, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (18, 23, s)-Nets in Base 81

(18, 23, large)-Net over F81 — Constructive and digital

(18, 23, large)-Net in Base 81 — Upper bound on s

(18, 23, large)-Net over F₈₁ — Constructive and digital