MinT - Best Known (51−7, 51, s)-Nets in Base 128

Best Known (51−7, 51, s)-Nets in Base 128

(51−7, 51, large)-Net over F₁₂₈ — Constructive and digital

Digital (44, 51, large)-net over F₁₂₈, using

128¹ times duplication [i] based on digital (43, 50, large)-net over F₁₂₈, using
- t-expansion [i] based on digital (41, 50, large)-net over F₁₂₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 524288)-net over F₁₂₈, using
      - s-reduction based on digital (0, 0, s)-net over F₁₂₈ with arbitrarily large s, using
        arbitrary set of 128⁰ points [i]
    2. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    3. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    4. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    5. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    6. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    7. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    8. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    9. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    10. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    11. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    12. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    13. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    14. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    15. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    16. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    17. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    18. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    19. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    20. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    21. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    22. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    23. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    24. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    25. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    26. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    27. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    28. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    29. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    30. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    31. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    32. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    33. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    34. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    35. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    36. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    37. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    38. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    39. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    40. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    41. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    42. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    43. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    44. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    45. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    46. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    47. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    48. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    49. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    50. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    51. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    52. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    53. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    54. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    55. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    56. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    57. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    58. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    59. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    60. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    61. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    62. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    63. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    64. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    65. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    66. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    67. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    68. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    69. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    70. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    71. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    72. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    73. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    74. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    75. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    76. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    77. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    78. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    79. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    80. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    81. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    82. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    83. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    84. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    85. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    86. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    87. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    88. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    89. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    90. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    91. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    92. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    93. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    94. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    95. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    96. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    97. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    98. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    99. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    100. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    101. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    102. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    103. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    104. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    105. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    106. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    107. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    108. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    109. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    110. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    111. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    112. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    113. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    114. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    115. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    116. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    117. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    118. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    119. digital (0, 0, 524288)-net over F₁₂₈ (see above)
    120. digital (0, 1, 524288)-net over F₁₂₈, using
      - s-reduction based on digital (0, 1, s)-net over F₁₂₈ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    121. digital (0, 1, 524288)-net over F₁₂₈ (see above)
    122. digital (0, 1, 524288)-net over F₁₂₈ (see above)
    123. digital (0, 1, 524288)-net over F₁₂₈ (see above)
    124. digital (0, 1, 524288)-net over F₁₂₈ (see above)
    125. digital (2, 4, 524288)-net over F₁₂₈, using
      - s-reduction based on digital (2, 4, 2113665)-net over F₁₂₈, using
        kÂ =Â 2 construction [i]
    126. digital (3, 6, 524288)-net over F₁₂₈, using
      - s-reduction based on digital (3, 6, 2130050)-net over F₁₂₈, using
        net defined by OOA [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 3, 3) (dual of [(2130050, 3), 6390144, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 2, 3) (dual of [(2130050, 2), 4260094, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁶, 2130050, F₁₂₈, 3) (dual of [2130050, 2130044, 4]-code or 2130050-cap in PG(5,128)), using
        large caps in PG(5,Â b) [i]
    127. digital (6, 10, 524288)-net over F₁₂₈, using
      - s-reduction based on digital (6, 10, 1048577)-net over F₁₂₈, using
        net defined by OOA [i] based on linear OOA(128¹⁰, 1048577, F₁₂₈, 4, 4) (dual of [(1048577, 4), 4194298, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(128¹⁰, 2097154, F₁₂₈, 4) (dual of [2097154, 2097144, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹⁰, 2097155, F₁₂₈, 4) (dual of [2097155, 2097145, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(128⁷, 2097152, F₁₂₈, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]
    128. digital (16, 25, 524288)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128²⁵, 524288, F₁₂₈, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OA(128²⁵, 2097153, F₁₂₈, 9) (dual of [2097153, 2097128, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(51−7, 51, large)-Net in Base 128 — Upper bound on s

There is no (44, 51, large)-net in base 128, because

5 times m-reduction [i] would yield (44, 46, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (51−7, 51, s)-Nets in Base 128

(51−7, 51, large)-Net over F128 — Constructive and digital

(51−7, 51, large)-Net in Base 128 — Upper bound on s

(51−7, 51, large)-Net over F₁₂₈ — Constructive and digital