MinT - Best Known (16, 16+4, s)-Nets in Base 128

Best Known (16, 16+4, s)-Nets in Base 128

(16, 16+4, large)-Net over F₁₂₈ — Constructive and digital

Digital (16, 20, large)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 20, large)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128²⁰, large, F₁₂₈, 5, 5), using
  - appending kth column [i] based on linear OOA(128²⁰, large, F₁₂₈, 4, 5), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁰, s, F₁₂₈, 4, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
      2. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      3. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      4. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      5. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      6. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      7. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      8. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      9. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      10. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      11. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      12. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      13. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      14. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      15. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      16. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      17. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      18. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      19. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      20. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      21. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      22. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      23. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      24. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      25. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      26. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      27. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      28. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      29. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      30. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      31. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      32. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      33. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      34. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      35. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      36. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      37. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      38. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      39. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      40. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      41. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      42. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      43. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      44. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      45. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      46. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      47. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      48. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      49. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      50. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      51. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      52. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      53. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      54. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      55. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      56. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      57. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      58. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      59. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      60. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      61. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      62. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      63. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      64. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      65. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      66. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      67. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      68. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      69. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      70. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      71. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      72. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      73. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      74. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      75. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      76. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      77. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      78. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      79. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      80. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      81. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      82. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      83. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      84. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      85. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      86. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      87. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      88. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      89. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      90. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      91. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      92. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      93. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      94. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      95. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      96. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      97. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      98. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      99. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      100. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      101. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      102. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      103. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      104. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      105. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      106. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      107. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      108. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      109. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      110. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      111. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      112. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      113. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      114. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      115. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      116. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      117. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      118. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      119. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      120. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      121. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      122. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      123. linear OOA(128⁰, 1048577, F₁₂₈, 4, 0) (dual of [(1048577, 4), 4194308, 1]-NRT-code) (see above)
      124. linear OOA(128¹, 1048577, F₁₂₈, 4, 1) (dual of [(1048577, 4), 4194307, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128¹, s, F₁₂₈, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      125. linear OOA(128¹, 1048577, F₁₂₈, 4, 1) (dual of [(1048577, 4), 4194307, 2]-NRT-code) (see above)
      126. linear OOA(128¹, 1048577, F₁₂₈, 4, 1) (dual of [(1048577, 4), 4194307, 2]-NRT-code) (see above)
      127. linear OOA(128⁴, 1048577, F₁₂₈, 4, 2) (dual of [(1048577, 4), 4194304, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁴, 2113665, F₁₂₈, 4, 2) (dual of [(2113665, 4), 8454656, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(128⁴, 2113665, F₁₂₈, 2, 2) (dual of [(2113665, 2), 4227326, 3]-NRT-code), using
        appending kth column [i] based on linear OA(128⁴, 2113665, F₁₂₈, 2) (dual of [2113665, 2113661, 3]-code), using
        Hamming code H(4,128) [i]
      128. linear OOA(128¹³, 1048577, F₁₂₈, 4, 5) (dual of [(1048577, 4), 4194295, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(128¹³, 2097155, F₁₂₈, 5) (dual of [2097155, 2097142, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(16, 16+4, large)-Net in Base 128 — Upper bound on s

There is no (16, 20, large)-net in base 128, because

2 times m-reduction [i] would yield (16, 18, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (16, 16+4, s)-Nets in Base 128

(16, 16+4, large)-Net over F128 — Constructive and digital

(16, 16+4, large)-Net in Base 128 — Upper bound on s

(16, 16+4, large)-Net over F₁₂₈ — Constructive and digital