MinT - Best Known (41, 41+8, s)-Nets in Base 128

Best Known (41, 41+8, s)-Nets in Base 128

(41, 41+8, large)-Net over F₁₂₈ — Constructive and digital

Digital (41, 49, large)-net over F₁₂₈, using

1 times m-reduction [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]

(41, 41+8, large)-Net in Base 128 — Upper bound on s

There is no (41, 49, large)-net in base 128, because

6 times m-reduction [i] would yield (41, 43, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (41, 41+8, s)-Nets in Base 128

(41, 41+8, large)-Net over F128 — Constructive and digital

(41, 41+8, large)-Net in Base 128 — Upper bound on s

(41, 41+8, large)-Net over F₁₂₈ — Constructive and digital