MinT - Best Known (16−7, 16, s)-Nets in Base 128

Best Known (16−7, 16, s)-Nets in Base 128

(16−7, 16, 16512)-Net over F₁₂₈ — Constructive and digital

Digital (9, 16, 16512)-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 (0, 3, 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 (0, 7, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(16−7, 16, 16516)-Net over F₁₂₈ — Digital

Digital (9, 16, 16516)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128¹⁶, 16516, F₁₂₈, 7) (dual of [16516, 16500, 8]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(128³, 130, F₁₂₈, 3) (dual of [130, 127, 4]-code or 130-arc in PG(2,128) or 130-cap in PG(2,128)), using
    - hyperoval in PG(2,Â 128) [i]
  2. linear OA(128¹³, 16386, F₁₂₈, 7) (dual of [16386, 16373, 8]-code), using
    - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
      1. linear OA(128¹³, 16384, F₁₂₈, 7) (dual of [16384, 16371, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OA(128¹¹, 16384, F₁₂₈, 6) (dual of [16384, 16373, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      3. linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 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]

(16−7, 16, 21847)-Net in Base 128 — Constructive

(9, 16, 21847)-net in base 128, using

base change [i] based on digital (7, 14, 21847)-net over F₂₅₆, using
- net defined by OOA [i] based on linear OOA(256¹⁴, 21847, F₂₅₆, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
  - appending kth column [i] based on linear OOA(256¹⁴, 21847, F₂₅₆, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(256¹⁴, 65542, F₂₅₆, 7) (dual of [65542, 65528, 8]-code), using
      - constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [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(256⁹, 65537, F₂₅₆, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(16−7, 16, 32771)-Net in Base 128

(9, 16, 32771)-net in base 128, using

base change [i] based on digital (7, 14, 32771)-net over F₂₅₆, using
- net defined by OOA [i] based on linear OOA(256¹⁴, 32771, F₂₅₆, 7, 7) (dual of [(32771, 7), 229383, 8]-NRT-code), using
  - appending kth column [i] based on linear OOA(256¹⁴, 32771, F₂₅₆, 6, 7) (dual of [(32771, 6), 196612, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256¹⁴, 32771, F₂₅₆, 2, 7) (dual of [(32771, 2), 65528, 8]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(256¹⁴, 65542, F₂₅₆, 7) (dual of [65542, 65528, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [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(256⁹, 65537, F₂₅₆, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(16−7, 16, large)-Net in Base 128 — Upper bound on s

There is no (9, 16, large)-net in base 128, because

5 times m-reduction [i] would yield (9, 11, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]