MinT - Best Known (50−9, 50, s)-Nets in Base 128

Best Known (50−9, 50, s)-Nets in Base 128

(50−9, 50, large)-Net over F₁₂₈ — Constructive and digital

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]

(50−9, 50, large)-Net in Base 128 — Upper bound on s

There is no (41, 50, large)-net in base 128, because

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

Best Known (50−9, 50, s)-Nets in Base 128

(50−9, 50, large)-Net over F128 — Constructive and digital

(50−9, 50, large)-Net in Base 128 — Upper bound on s

(50−9, 50, large)-Net over F₁₂₈ — Constructive and digital