MinT - Best Known (11, 18, s)-Nets in Base 128

Best Known (11, 18, s)-Nets in Base 128

(11, 18, 16554)-Net over F₁₂₈ — Constructive and digital

Digital (11, 18, 16554)-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, 0, 129)-net over F₁₂₈ (see above)
122. 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]
123. digital (0, 1, 129)-net over F₁₂₈ (see above)
124. digital (0, 1, 129)-net over F₁₂₈ (see above)
125. digital (0, 1, 129)-net over F₁₂₈ (see above)
126. digital (0, 2, 129)-net over F₁₂₈, using
  - kÂ =Â 2 construction [i]
127. digital (1, 4, 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]
128. digital (1, 8, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(11, 18, 21974)-Net in Base 128 — Constructive

(11, 18, 21974)-net in base 128, using

net defined by OOA [i] based on OOA(128¹⁸, 21974, S₁₂₈, 9, 7), using
- OOA stacking with additional row [i] based on OOA(128¹⁸, 21975, S₁₂₈, 3, 7), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(128³, 129, F₁₂₈, 3, 3) (dual of [(129, 3), 384, 4]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(3;384,128) [i]
    2. OOA(128¹⁵, 21846, S₁₂₈, 3, 7), using
      - OOA 3-folding [i] based on OA(128¹⁵, 65538, S₁₂₈, 7), using
        discarding parts of the base [i] based on linear OA(256¹³, 65538, F₂₅₆, 7) (dual of [65538, 65525, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(256¹¹, 65536, F₂₅₆, 6) (dual of [65536, 65525, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

(11, 18, 49440)-Net over F₁₂₈ — Digital

Digital (11, 18, 49440)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(11, 18, 56539)-Net in Base 128

(11, 18, 56539)-net in base 128, using

net defined by OOA [i] based on OOA(128¹⁸, 56539, S₁₂₈, 9, 7), using
- OOA stacking with additional row [i] based on OOA(128¹⁸, 56540, S₁₂₈, 3, 7), using
  - discarding parts of the base [i] based on linear OOA(256¹⁵, 56540, F₂₅₆, 3, 7) (dual of [(56540, 3), 169605, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256¹⁵, 56540, F₂₅₆, 7) (dual of [56540, 56525, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(256¹⁵, 65540, F₂₅₆, 7) (dual of [65540, 65525, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([1,3]) [i] based on
        linear OA(256¹³, 65537, F₂₅₆, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(256¹², 65537, F₂₅₆, 4) (dual of [65537, 65525, 5]-code), using the narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [1,3], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]
        linear OA(256², 3, F₂₅₆, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,256)), using
        dual of repetition code with length 3 [i]

(11, 18, large)-Net in Base 128 — Upper bound on s

There is no (11, 18, large)-net in base 128, because

5 times m-reduction [i] would yield (11, 13, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]