MinT - Best Known (9−4, 9, s)-Nets in Base 128

Best Known (9−4, 9, s)-Nets in Base 128

(9−4, 9, 16533)-Net over F₁₂₈ — Constructive and digital

Digital (5, 9, 16533)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128⁹, 16533, F₁₂₈, 4, 4) (dual of [(16533, 4), 66123, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(128⁹, 16533, F₁₂₈, 3, 4) (dual of [(16533, 3), 49590, 5]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(128⁰, s, F₁₂₈, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    3. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    4. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    5. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    6. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    7. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    8. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    9. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    10. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    11. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    12. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    13. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    14. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    15. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    16. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    17. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    18. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    19. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    20. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    21. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    22. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    23. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    24. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    25. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    26. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    27. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    28. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    29. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    30. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    31. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    32. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    33. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    34. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    35. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    36. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    37. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    38. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    39. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    40. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    41. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    42. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    43. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    44. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    45. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    46. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    47. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    48. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    49. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    50. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    51. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    52. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    53. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    54. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    55. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    56. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    57. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    58. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    59. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    60. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    61. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    62. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    63. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    64. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    65. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    66. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    67. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    68. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    69. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    70. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    71. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    72. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    73. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    74. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    75. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    76. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    77. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    78. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    79. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    80. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    81. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    82. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    83. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    84. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    85. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    86. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    87. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    88. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    89. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    90. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    91. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    92. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    93. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    94. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    95. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    96. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    97. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    98. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    99. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    100. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    101. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    102. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    103. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    104. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    105. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    106. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    107. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    108. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    109. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    110. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    111. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    112. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    113. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    114. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    115. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    116. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    117. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    118. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    119. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    120. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    121. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    122. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    123. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    124. linear OOA(128⁰, 129, F₁₂₈, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
    125. linear OOA(128¹, 129, F₁₂₈, 3, 1) (dual of [(129, 3), 386, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(128¹, s, F₁₂₈, 3, 1) with arbitrarily large s, using
        appending 2 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]
    126. linear OOA(128¹, 129, F₁₂₈, 3, 1) (dual of [(129, 3), 386, 2]-NRT-code) (see above)
    127. linear OOA(128², 129, F₁₂₈, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(3;385,128) [i]
    128. linear OOA(128⁵, 150, F₁₂₈, 3, 4) (dual of [(150, 3), 445, 5]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(3;F,445P) [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(9−4, 9, 32513)-Net over F₁₂₈ — Digital

Digital (5, 9, 32513)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128⁹, 32513, F₁₂₈, 4, 4) (dual of [(32513, 4), 130043, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(128⁹, 32513, F₁₂₈, 3, 4) (dual of [(32513, 3), 97530, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁹, 32513, F₁₂₈, 4) (dual of [32513, 32504, 5]-code), using
    - 1 times truncation [i] based on linear OA(128¹⁰, 32514, F₁₂₈, 5) (dual of [32514, 32504, 6]-code), using
      - (u,Â u+v)-construction [i] based on
        linear OA(128³, 16257, F₁₂₈, 2) (dual of [16257, 16254, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(128³, 16513, F₁₂₈, 2) (dual of [16513, 16510, 3]-code), using
        Hamming code H(3,128) [i]
        linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
        a code by Danev and Olsson [i]

(9−4, 9, 32769)-Net in Base 128 — Constructive

(5, 9, 32769)-net in base 128, using

1 times m-reduction [i] based on (5, 10, 32769)-net in base 128, using
- net defined by OOA [i] based on OOA(128¹⁰, 32769, S₁₂₈, 5, 5), using
  - appending kth column [i] based on OOA(128¹⁰, 32769, S₁₂₈, 4, 5), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(128², 129, F₁₂₈, 4, 2) (dual of [(129, 4), 514, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;514,128) [i]
      2. OOA(128⁸, 32640, S₁₂₈, 4, 5), using
        OOA 2-folding and stacking with additional row [i] based on OA(128⁸, 65281, S₁₂₈, 5), using
        discarding parts of the base [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]

(9−4, 9, 65538)-Net in Base 128

(5, 9, 65538)-net in base 128, using

128¹ times duplication [i] based on (4, 8, 65538)-net in base 128, using
- base change [i] based on digital (3, 7, 65538)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁷, 65538, F₂₅₆, 4, 4) (dual of [(65538, 4), 262145, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(256⁷, 65538, F₂₅₆, 3, 4) (dual of [(65538, 3), 196607, 5]-NRT-code), using
      - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁷, 65538, F₂₅₆, 4) (dual of [65538, 65531, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(256⁷, 65536, F₂₅₆, 4) (dual of [65536, 65529, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(256⁵, 65536, F₂₅₆, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(9−4, 9, large)-Net in Base 128 — Upper bound on s

There is no (5, 9, large)-net in base 128, because

2 times m-reduction [i] would yield (5, 7, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]