MinT - Best Known (13, 21, s)-Nets in Base 128

Best Known (13, 21, s)-Nets in Base 128

(13, 21, 16533)-Net over F₁₂₈ — Constructive and digital

Digital (13, 21, 16533)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 129)-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, 129)-net over F₁₂₈ (see above)
3. digital (0, 0, 129)-net over F₁₂₈ (see above)
4. digital (0, 0, 129)-net over F₁₂₈ (see above)
5. digital (0, 0, 129)-net over F₁₂₈ (see above)
6. digital (0, 0, 129)-net over F₁₂₈ (see above)
7. digital (0, 0, 129)-net over F₁₂₈ (see above)
8. digital (0, 0, 129)-net over F₁₂₈ (see above)
9. digital (0, 0, 129)-net over F₁₂₈ (see above)
10. digital (0, 0, 129)-net over F₁₂₈ (see above)
11. digital (0, 0, 129)-net over F₁₂₈ (see above)
12. digital (0, 0, 129)-net over F₁₂₈ (see above)
13. digital (0, 0, 129)-net over F₁₂₈ (see above)
14. digital (0, 0, 129)-net over F₁₂₈ (see above)
15. digital (0, 0, 129)-net over F₁₂₈ (see above)
16. digital (0, 0, 129)-net over F₁₂₈ (see above)
17. digital (0, 0, 129)-net over F₁₂₈ (see above)
18. digital (0, 0, 129)-net over F₁₂₈ (see above)
19. digital (0, 0, 129)-net over F₁₂₈ (see above)
20. digital (0, 0, 129)-net over F₁₂₈ (see above)
21. digital (0, 0, 129)-net over F₁₂₈ (see above)
22. digital (0, 0, 129)-net over F₁₂₈ (see above)
23. digital (0, 0, 129)-net over F₁₂₈ (see above)
24. digital (0, 0, 129)-net over F₁₂₈ (see above)
25. digital (0, 0, 129)-net over F₁₂₈ (see above)
26. digital (0, 0, 129)-net over F₁₂₈ (see above)
27. digital (0, 0, 129)-net over F₁₂₈ (see above)
28. digital (0, 0, 129)-net over F₁₂₈ (see above)
29. digital (0, 0, 129)-net over F₁₂₈ (see above)
30. digital (0, 0, 129)-net over F₁₂₈ (see above)
31. digital (0, 0, 129)-net over F₁₂₈ (see above)
32. digital (0, 0, 129)-net over F₁₂₈ (see above)
33. digital (0, 0, 129)-net over F₁₂₈ (see above)
34. digital (0, 0, 129)-net over F₁₂₈ (see above)
35. digital (0, 0, 129)-net over F₁₂₈ (see above)
36. digital (0, 0, 129)-net over F₁₂₈ (see above)
37. digital (0, 0, 129)-net over F₁₂₈ (see above)
38. digital (0, 0, 129)-net over F₁₂₈ (see above)
39. digital (0, 0, 129)-net over F₁₂₈ (see above)
40. digital (0, 0, 129)-net over F₁₂₈ (see above)
41. digital (0, 0, 129)-net over F₁₂₈ (see above)
42. digital (0, 0, 129)-net over F₁₂₈ (see above)
43. digital (0, 0, 129)-net over F₁₂₈ (see above)
44. digital (0, 0, 129)-net over F₁₂₈ (see above)
45. digital (0, 0, 129)-net over F₁₂₈ (see above)
46. digital (0, 0, 129)-net over F₁₂₈ (see above)
47. digital (0, 0, 129)-net over F₁₂₈ (see above)
48. digital (0, 0, 129)-net over F₁₂₈ (see above)
49. digital (0, 0, 129)-net over F₁₂₈ (see above)
50. digital (0, 0, 129)-net over F₁₂₈ (see above)
51. digital (0, 0, 129)-net over F₁₂₈ (see above)
52. digital (0, 0, 129)-net over F₁₂₈ (see above)
53. digital (0, 0, 129)-net over F₁₂₈ (see above)
54. digital (0, 0, 129)-net over F₁₂₈ (see above)
55. digital (0, 0, 129)-net over F₁₂₈ (see above)
56. digital (0, 0, 129)-net over F₁₂₈ (see above)
57. digital (0, 0, 129)-net over F₁₂₈ (see above)
58. digital (0, 0, 129)-net over F₁₂₈ (see above)
59. digital (0, 0, 129)-net over F₁₂₈ (see above)
60. digital (0, 0, 129)-net over F₁₂₈ (see above)
61. digital (0, 0, 129)-net over F₁₂₈ (see above)
62. digital (0, 0, 129)-net over F₁₂₈ (see above)
63. digital (0, 0, 129)-net over F₁₂₈ (see above)
64. digital (0, 0, 129)-net over F₁₂₈ (see above)
65. digital (0, 0, 129)-net over F₁₂₈ (see above)
66. digital (0, 0, 129)-net over F₁₂₈ (see above)
67. digital (0, 0, 129)-net over F₁₂₈ (see above)
68. digital (0, 0, 129)-net over F₁₂₈ (see above)
69. digital (0, 0, 129)-net over F₁₂₈ (see above)
70. digital (0, 0, 129)-net over F₁₂₈ (see above)
71. digital (0, 0, 129)-net over F₁₂₈ (see above)
72. digital (0, 0, 129)-net over F₁₂₈ (see above)
73. digital (0, 0, 129)-net over F₁₂₈ (see above)
74. digital (0, 0, 129)-net over F₁₂₈ (see above)
75. digital (0, 0, 129)-net over F₁₂₈ (see above)
76. digital (0, 0, 129)-net over F₁₂₈ (see above)
77. digital (0, 0, 129)-net over F₁₂₈ (see above)
78. digital (0, 0, 129)-net over F₁₂₈ (see above)
79. digital (0, 0, 129)-net over F₁₂₈ (see above)
80. digital (0, 0, 129)-net over F₁₂₈ (see above)
81. digital (0, 0, 129)-net over F₁₂₈ (see above)
82. digital (0, 0, 129)-net over F₁₂₈ (see above)
83. digital (0, 0, 129)-net over F₁₂₈ (see above)
84. digital (0, 0, 129)-net over F₁₂₈ (see above)
85. digital (0, 0, 129)-net over F₁₂₈ (see above)
86. digital (0, 0, 129)-net over F₁₂₈ (see above)
87. digital (0, 0, 129)-net over F₁₂₈ (see above)
88. digital (0, 0, 129)-net over F₁₂₈ (see above)
89. digital (0, 0, 129)-net over F₁₂₈ (see above)
90. digital (0, 0, 129)-net over F₁₂₈ (see above)
91. digital (0, 0, 129)-net over F₁₂₈ (see above)
92. digital (0, 0, 129)-net over F₁₂₈ (see above)
93. digital (0, 0, 129)-net over F₁₂₈ (see above)
94. digital (0, 0, 129)-net over F₁₂₈ (see above)
95. digital (0, 0, 129)-net over F₁₂₈ (see above)
96. digital (0, 0, 129)-net over F₁₂₈ (see above)
97. digital (0, 0, 129)-net over F₁₂₈ (see above)
98. digital (0, 0, 129)-net over F₁₂₈ (see above)
99. digital (0, 0, 129)-net over F₁₂₈ (see above)
100. digital (0, 0, 129)-net over F₁₂₈ (see above)
101. digital (0, 0, 129)-net over F₁₂₈ (see above)
102. digital (0, 0, 129)-net over F₁₂₈ (see above)
103. digital (0, 0, 129)-net over F₁₂₈ (see above)
104. digital (0, 0, 129)-net over F₁₂₈ (see above)
105. digital (0, 0, 129)-net over F₁₂₈ (see above)
106. digital (0, 0, 129)-net over F₁₂₈ (see above)
107. digital (0, 0, 129)-net over F₁₂₈ (see above)
108. digital (0, 0, 129)-net over F₁₂₈ (see above)
109. digital (0, 0, 129)-net over F₁₂₈ (see above)
110. digital (0, 0, 129)-net over F₁₂₈ (see above)
111. digital (0, 0, 129)-net over F₁₂₈ (see above)
112. digital (0, 0, 129)-net over F₁₂₈ (see above)
113. digital (0, 0, 129)-net over F₁₂₈ (see above)
114. digital (0, 0, 129)-net over F₁₂₈ (see above)
115. digital (0, 0, 129)-net over F₁₂₈ (see above)
116. digital (0, 0, 129)-net over F₁₂₈ (see above)
117. digital (0, 0, 129)-net over F₁₂₈ (see above)
118. digital (0, 0, 129)-net over F₁₂₈ (see above)
119. digital (0, 0, 129)-net over F₁₂₈ (see above)
120. digital (0, 0, 129)-net over F₁₂₈ (see above)
121. digital (0, 1, 129)-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]
122. digital (0, 1, 129)-net over F₁₂₈ (see above)
123. digital (0, 1, 129)-net over F₁₂₈ (see above)
124. digital (0, 1, 129)-net over F₁₂₈ (see above)
125. digital (0, 2, 129)-net over F₁₂₈, using
  - kÂ =Â 2 construction [i]
126. digital (0, 2, 129)-net over F₁₂₈ (see above)
127. digital (0, 4, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
128. digital (1, 9, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(13, 21, 55818)-Net over F₁₂₈ — Digital

Digital (13, 21, 55818)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(13, 21, 65546)-Net in Base 128

(13, 21, 65546)-net in base 128, using

net defined by OOA [i] based on OOA(128²¹, 65546, S₁₂₈, 12, 8), using
- OOA stacking with additional row [i] based on OOA(128²¹, 65547, S₁₂₈, 4, 8), using
  - discarding parts of the base [i] based on linear OOA(256¹⁸, 65547, F₂₅₆, 4, 8) (dual of [(65547, 4), 262170, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256¹⁸, 65547, F₂₅₆, 8) (dual of [65547, 65529, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(3) [i] based on
        linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        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(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(13, 21, large)-Net in Base 128 — Upper bound on s

There is no (13, 21, large)-net in base 128, because

6 times m-reduction [i] would yield (13, 15, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (13, 21, s)-Nets in Base 128

(13, 21, 16533)-Net over F128 — Constructive and digital

(13, 21, 55818)-Net over F128 — Digital

(13, 21, 65546)-Net in Base 128

(13, 21, large)-Net in Base 128 — Upper bound on s

(13, 21, 16533)-Net over F₁₂₈ — Constructive and digital

(13, 21, 55818)-Net over F₁₂₈ — Digital