MinT - Best Known (42−10, 42, s)-Nets in Base 256

Best Known (42−10, 42, s)-Nets in Base 256

(42−10, 42, 3355648)-Net over F₂₅₆ — Constructive and digital

Digital (32, 42, 3355648)-net over F₂₅₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 13108)-net over F₂₅₆, using
  - s-reduction based on digital (0, 0, s)-net over F₂₅₆ with arbitrarily large s, using
    - arbitrary set of 256⁰ points [i]
2. digital (0, 0, 13108)-net over F₂₅₆ (see above)
3. digital (0, 0, 13108)-net over F₂₅₆ (see above)
4. digital (0, 0, 13108)-net over F₂₅₆ (see above)
5. digital (0, 0, 13108)-net over F₂₅₆ (see above)
6. digital (0, 0, 13108)-net over F₂₅₆ (see above)
7. digital (0, 0, 13108)-net over F₂₅₆ (see above)
8. digital (0, 0, 13108)-net over F₂₅₆ (see above)
9. digital (0, 0, 13108)-net over F₂₅₆ (see above)
10. digital (0, 0, 13108)-net over F₂₅₆ (see above)
11. digital (0, 0, 13108)-net over F₂₅₆ (see above)
12. digital (0, 0, 13108)-net over F₂₅₆ (see above)
13. digital (0, 0, 13108)-net over F₂₅₆ (see above)
14. digital (0, 0, 13108)-net over F₂₅₆ (see above)
15. digital (0, 0, 13108)-net over F₂₅₆ (see above)
16. digital (0, 0, 13108)-net over F₂₅₆ (see above)
17. digital (0, 0, 13108)-net over F₂₅₆ (see above)
18. digital (0, 0, 13108)-net over F₂₅₆ (see above)
19. digital (0, 0, 13108)-net over F₂₅₆ (see above)
20. digital (0, 0, 13108)-net over F₂₅₆ (see above)
21. digital (0, 0, 13108)-net over F₂₅₆ (see above)
22. digital (0, 0, 13108)-net over F₂₅₆ (see above)
23. digital (0, 0, 13108)-net over F₂₅₆ (see above)
24. digital (0, 0, 13108)-net over F₂₅₆ (see above)
25. digital (0, 0, 13108)-net over F₂₅₆ (see above)
26. digital (0, 0, 13108)-net over F₂₅₆ (see above)
27. digital (0, 0, 13108)-net over F₂₅₆ (see above)
28. digital (0, 0, 13108)-net over F₂₅₆ (see above)
29. digital (0, 0, 13108)-net over F₂₅₆ (see above)
30. digital (0, 0, 13108)-net over F₂₅₆ (see above)
31. digital (0, 0, 13108)-net over F₂₅₆ (see above)
32. digital (0, 0, 13108)-net over F₂₅₆ (see above)
33. digital (0, 0, 13108)-net over F₂₅₆ (see above)
34. digital (0, 0, 13108)-net over F₂₅₆ (see above)
35. digital (0, 0, 13108)-net over F₂₅₆ (see above)
36. digital (0, 0, 13108)-net over F₂₅₆ (see above)
37. digital (0, 0, 13108)-net over F₂₅₆ (see above)
38. digital (0, 0, 13108)-net over F₂₅₆ (see above)
39. digital (0, 0, 13108)-net over F₂₅₆ (see above)
40. digital (0, 0, 13108)-net over F₂₅₆ (see above)
41. digital (0, 0, 13108)-net over F₂₅₆ (see above)
42. digital (0, 0, 13108)-net over F₂₅₆ (see above)
43. digital (0, 0, 13108)-net over F₂₅₆ (see above)
44. digital (0, 0, 13108)-net over F₂₅₆ (see above)
45. digital (0, 0, 13108)-net over F₂₅₆ (see above)
46. digital (0, 0, 13108)-net over F₂₅₆ (see above)
47. digital (0, 0, 13108)-net over F₂₅₆ (see above)
48. digital (0, 0, 13108)-net over F₂₅₆ (see above)
49. digital (0, 0, 13108)-net over F₂₅₆ (see above)
50. digital (0, 0, 13108)-net over F₂₅₆ (see above)
51. digital (0, 0, 13108)-net over F₂₅₆ (see above)
52. digital (0, 0, 13108)-net over F₂₅₆ (see above)
53. digital (0, 0, 13108)-net over F₂₅₆ (see above)
54. digital (0, 0, 13108)-net over F₂₅₆ (see above)
55. digital (0, 0, 13108)-net over F₂₅₆ (see above)
56. digital (0, 0, 13108)-net over F₂₅₆ (see above)
57. digital (0, 0, 13108)-net over F₂₅₆ (see above)
58. digital (0, 0, 13108)-net over F₂₅₆ (see above)
59. digital (0, 0, 13108)-net over F₂₅₆ (see above)
60. digital (0, 0, 13108)-net over F₂₅₆ (see above)
61. digital (0, 0, 13108)-net over F₂₅₆ (see above)
62. digital (0, 0, 13108)-net over F₂₅₆ (see above)
63. digital (0, 0, 13108)-net over F₂₅₆ (see above)
64. digital (0, 0, 13108)-net over F₂₅₆ (see above)
65. digital (0, 0, 13108)-net over F₂₅₆ (see above)
66. digital (0, 0, 13108)-net over F₂₅₆ (see above)
67. digital (0, 0, 13108)-net over F₂₅₆ (see above)
68. digital (0, 0, 13108)-net over F₂₅₆ (see above)
69. digital (0, 0, 13108)-net over F₂₅₆ (see above)
70. digital (0, 0, 13108)-net over F₂₅₆ (see above)
71. digital (0, 0, 13108)-net over F₂₅₆ (see above)
72. digital (0, 0, 13108)-net over F₂₅₆ (see above)
73. digital (0, 0, 13108)-net over F₂₅₆ (see above)
74. digital (0, 0, 13108)-net over F₂₅₆ (see above)
75. digital (0, 0, 13108)-net over F₂₅₆ (see above)
76. digital (0, 0, 13108)-net over F₂₅₆ (see above)
77. digital (0, 0, 13108)-net over F₂₅₆ (see above)
78. digital (0, 0, 13108)-net over F₂₅₆ (see above)
79. digital (0, 0, 13108)-net over F₂₅₆ (see above)
80. digital (0, 0, 13108)-net over F₂₅₆ (see above)
81. digital (0, 0, 13108)-net over F₂₅₆ (see above)
82. digital (0, 0, 13108)-net over F₂₅₆ (see above)
83. digital (0, 0, 13108)-net over F₂₅₆ (see above)
84. digital (0, 0, 13108)-net over F₂₅₆ (see above)
85. digital (0, 0, 13108)-net over F₂₅₆ (see above)
86. digital (0, 0, 13108)-net over F₂₅₆ (see above)
87. digital (0, 0, 13108)-net over F₂₅₆ (see above)
88. digital (0, 0, 13108)-net over F₂₅₆ (see above)
89. digital (0, 0, 13108)-net over F₂₅₆ (see above)
90. digital (0, 0, 13108)-net over F₂₅₆ (see above)
91. digital (0, 0, 13108)-net over F₂₅₆ (see above)
92. digital (0, 0, 13108)-net over F₂₅₆ (see above)
93. digital (0, 0, 13108)-net over F₂₅₆ (see above)
94. digital (0, 0, 13108)-net over F₂₅₆ (see above)
95. digital (0, 0, 13108)-net over F₂₅₆ (see above)
96. digital (0, 0, 13108)-net over F₂₅₆ (see above)
97. digital (0, 0, 13108)-net over F₂₅₆ (see above)
98. digital (0, 0, 13108)-net over F₂₅₆ (see above)
99. digital (0, 0, 13108)-net over F₂₅₆ (see above)
100. digital (0, 0, 13108)-net over F₂₅₆ (see above)
101. digital (0, 0, 13108)-net over F₂₅₆ (see above)
102. digital (0, 0, 13108)-net over F₂₅₆ (see above)
103. digital (0, 0, 13108)-net over F₂₅₆ (see above)
104. digital (0, 0, 13108)-net over F₂₅₆ (see above)
105. digital (0, 0, 13108)-net over F₂₅₆ (see above)
106. digital (0, 0, 13108)-net over F₂₅₆ (see above)
107. digital (0, 0, 13108)-net over F₂₅₆ (see above)
108. digital (0, 0, 13108)-net over F₂₅₆ (see above)
109. digital (0, 0, 13108)-net over F₂₅₆ (see above)
110. digital (0, 0, 13108)-net over F₂₅₆ (see above)
111. digital (0, 0, 13108)-net over F₂₅₆ (see above)
112. digital (0, 0, 13108)-net over F₂₅₆ (see above)
113. digital (0, 0, 13108)-net over F₂₅₆ (see above)
114. digital (0, 0, 13108)-net over F₂₅₆ (see above)
115. digital (0, 0, 13108)-net over F₂₅₆ (see above)
116. digital (0, 0, 13108)-net over F₂₅₆ (see above)
117. digital (0, 0, 13108)-net over F₂₅₆ (see above)
118. digital (0, 0, 13108)-net over F₂₅₆ (see above)
119. digital (0, 0, 13108)-net over F₂₅₆ (see above)
120. digital (0, 0, 13108)-net over F₂₅₆ (see above)
121. digital (0, 0, 13108)-net over F₂₅₆ (see above)
122. digital (0, 0, 13108)-net over F₂₅₆ (see above)
123. digital (0, 0, 13108)-net over F₂₅₆ (see above)
124. digital (0, 0, 13108)-net over F₂₅₆ (see above)
125. digital (0, 0, 13108)-net over F₂₅₆ (see above)
126. digital (0, 0, 13108)-net over F₂₅₆ (see above)
127. digital (0, 0, 13108)-net over F₂₅₆ (see above)
128. digital (0, 0, 13108)-net over F₂₅₆ (see above)
129. digital (0, 0, 13108)-net over F₂₅₆ (see above)
130. digital (0, 0, 13108)-net over F₂₅₆ (see above)
131. digital (0, 0, 13108)-net over F₂₅₆ (see above)
132. digital (0, 0, 13108)-net over F₂₅₆ (see above)
133. digital (0, 0, 13108)-net over F₂₅₆ (see above)
134. digital (0, 0, 13108)-net over F₂₅₆ (see above)
135. digital (0, 0, 13108)-net over F₂₅₆ (see above)
136. digital (0, 0, 13108)-net over F₂₅₆ (see above)
137. digital (0, 0, 13108)-net over F₂₅₆ (see above)
138. digital (0, 0, 13108)-net over F₂₅₆ (see above)
139. digital (0, 0, 13108)-net over F₂₅₆ (see above)
140. digital (0, 0, 13108)-net over F₂₅₆ (see above)
141. digital (0, 0, 13108)-net over F₂₅₆ (see above)
142. digital (0, 0, 13108)-net over F₂₅₆ (see above)
143. digital (0, 0, 13108)-net over F₂₅₆ (see above)
144. digital (0, 0, 13108)-net over F₂₅₆ (see above)
145. digital (0, 0, 13108)-net over F₂₅₆ (see above)
146. digital (0, 0, 13108)-net over F₂₅₆ (see above)
147. digital (0, 0, 13108)-net over F₂₅₆ (see above)
148. digital (0, 0, 13108)-net over F₂₅₆ (see above)
149. digital (0, 0, 13108)-net over F₂₅₆ (see above)
150. digital (0, 0, 13108)-net over F₂₅₆ (see above)
151. digital (0, 0, 13108)-net over F₂₅₆ (see above)
152. digital (0, 0, 13108)-net over F₂₅₆ (see above)
153. digital (0, 0, 13108)-net over F₂₅₆ (see above)
154. digital (0, 0, 13108)-net over F₂₅₆ (see above)
155. digital (0, 0, 13108)-net over F₂₅₆ (see above)
156. digital (0, 0, 13108)-net over F₂₅₆ (see above)
157. digital (0, 0, 13108)-net over F₂₅₆ (see above)
158. digital (0, 0, 13108)-net over F₂₅₆ (see above)
159. digital (0, 0, 13108)-net over F₂₅₆ (see above)
160. digital (0, 0, 13108)-net over F₂₅₆ (see above)
161. digital (0, 0, 13108)-net over F₂₅₆ (see above)
162. digital (0, 0, 13108)-net over F₂₅₆ (see above)
163. digital (0, 0, 13108)-net over F₂₅₆ (see above)
164. digital (0, 0, 13108)-net over F₂₅₆ (see above)
165. digital (0, 0, 13108)-net over F₂₅₆ (see above)
166. digital (0, 0, 13108)-net over F₂₅₆ (see above)
167. digital (0, 0, 13108)-net over F₂₅₆ (see above)
168. digital (0, 0, 13108)-net over F₂₅₆ (see above)
169. digital (0, 0, 13108)-net over F₂₅₆ (see above)
170. digital (0, 0, 13108)-net over F₂₅₆ (see above)
171. digital (0, 0, 13108)-net over F₂₅₆ (see above)
172. digital (0, 0, 13108)-net over F₂₅₆ (see above)
173. digital (0, 0, 13108)-net over F₂₅₆ (see above)
174. digital (0, 0, 13108)-net over F₂₅₆ (see above)
175. digital (0, 0, 13108)-net over F₂₅₆ (see above)
176. digital (0, 0, 13108)-net over F₂₅₆ (see above)
177. digital (0, 0, 13108)-net over F₂₅₆ (see above)
178. digital (0, 0, 13108)-net over F₂₅₆ (see above)
179. digital (0, 0, 13108)-net over F₂₅₆ (see above)
180. digital (0, 0, 13108)-net over F₂₅₆ (see above)
181. digital (0, 0, 13108)-net over F₂₅₆ (see above)
182. digital (0, 0, 13108)-net over F₂₅₆ (see above)
183. digital (0, 0, 13108)-net over F₂₅₆ (see above)
184. digital (0, 0, 13108)-net over F₂₅₆ (see above)
185. digital (0, 0, 13108)-net over F₂₅₆ (see above)
186. digital (0, 0, 13108)-net over F₂₅₆ (see above)
187. digital (0, 0, 13108)-net over F₂₅₆ (see above)
188. digital (0, 0, 13108)-net over F₂₅₆ (see above)
189. digital (0, 0, 13108)-net over F₂₅₆ (see above)
190. digital (0, 0, 13108)-net over F₂₅₆ (see above)
191. digital (0, 0, 13108)-net over F₂₅₆ (see above)
192. digital (0, 0, 13108)-net over F₂₅₆ (see above)
193. digital (0, 0, 13108)-net over F₂₅₆ (see above)
194. digital (0, 0, 13108)-net over F₂₅₆ (see above)
195. digital (0, 0, 13108)-net over F₂₅₆ (see above)
196. digital (0, 0, 13108)-net over F₂₅₆ (see above)
197. digital (0, 0, 13108)-net over F₂₅₆ (see above)
198. digital (0, 0, 13108)-net over F₂₅₆ (see above)
199. digital (0, 0, 13108)-net over F₂₅₆ (see above)
200. digital (0, 0, 13108)-net over F₂₅₆ (see above)
201. digital (0, 0, 13108)-net over F₂₅₆ (see above)
202. digital (0, 0, 13108)-net over F₂₅₆ (see above)
203. digital (0, 0, 13108)-net over F₂₅₆ (see above)
204. digital (0, 0, 13108)-net over F₂₅₆ (see above)
205. digital (0, 0, 13108)-net over F₂₅₆ (see above)
206. digital (0, 0, 13108)-net over F₂₅₆ (see above)
207. digital (0, 0, 13108)-net over F₂₅₆ (see above)
208. digital (0, 0, 13108)-net over F₂₅₆ (see above)
209. digital (0, 0, 13108)-net over F₂₅₆ (see above)
210. digital (0, 0, 13108)-net over F₂₅₆ (see above)
211. digital (0, 0, 13108)-net over F₂₅₆ (see above)
212. digital (0, 0, 13108)-net over F₂₅₆ (see above)
213. digital (0, 0, 13108)-net over F₂₅₆ (see above)
214. digital (0, 0, 13108)-net over F₂₅₆ (see above)
215. digital (0, 0, 13108)-net over F₂₅₆ (see above)
216. digital (0, 0, 13108)-net over F₂₅₆ (see above)
217. digital (0, 0, 13108)-net over F₂₅₆ (see above)
218. digital (0, 0, 13108)-net over F₂₅₆ (see above)
219. digital (0, 0, 13108)-net over F₂₅₆ (see above)
220. digital (0, 0, 13108)-net over F₂₅₆ (see above)
221. digital (0, 0, 13108)-net over F₂₅₆ (see above)
222. digital (0, 0, 13108)-net over F₂₅₆ (see above)
223. digital (0, 0, 13108)-net over F₂₅₆ (see above)
224. digital (0, 0, 13108)-net over F₂₅₆ (see above)
225. digital (0, 0, 13108)-net over F₂₅₆ (see above)
226. digital (0, 0, 13108)-net over F₂₅₆ (see above)
227. digital (0, 0, 13108)-net over F₂₅₆ (see above)
228. digital (0, 0, 13108)-net over F₂₅₆ (see above)
229. digital (0, 0, 13108)-net over F₂₅₆ (see above)
230. digital (0, 0, 13108)-net over F₂₅₆ (see above)
231. digital (0, 0, 13108)-net over F₂₅₆ (see above)
232. digital (0, 0, 13108)-net over F₂₅₆ (see above)
233. digital (0, 0, 13108)-net over F₂₅₆ (see above)
234. digital (0, 0, 13108)-net over F₂₅₆ (see above)
235. digital (0, 0, 13108)-net over F₂₅₆ (see above)
236. digital (0, 0, 13108)-net over F₂₅₆ (see above)
237. digital (0, 0, 13108)-net over F₂₅₆ (see above)
238. digital (0, 0, 13108)-net over F₂₅₆ (see above)
239. digital (0, 0, 13108)-net over F₂₅₆ (see above)
240. digital (0, 0, 13108)-net over F₂₅₆ (see above)
241. digital (0, 0, 13108)-net over F₂₅₆ (see above)
242. digital (0, 0, 13108)-net over F₂₅₆ (see above)
243. digital (0, 0, 13108)-net over F₂₅₆ (see above)
244. digital (0, 0, 13108)-net over F₂₅₆ (see above)
245. digital (0, 0, 13108)-net over F₂₅₆ (see above)
246. digital (0, 0, 13108)-net over F₂₅₆ (see above)
247. digital (0, 1, 13108)-net over F₂₅₆, using
  - s-reduction based on digital (0, 1, s)-net over F₂₅₆ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
248. digital (0, 1, 13108)-net over F₂₅₆ (see above)
249. digital (0, 1, 13108)-net over F₂₅₆ (see above)
250. digital (0, 1, 13108)-net over F₂₅₆ (see above)
251. digital (0, 1, 13108)-net over F₂₅₆ (see above)
252. digital (1, 3, 13108)-net over F₂₅₆, using
  - s-reduction based on digital (1, 3, 65793)-net over F₂₅₆, using
    - kÂ =Â 2 construction [i]
253. digital (1, 3, 13108)-net over F₂₅₆ (see above)
254. digital (1, 4, 13108)-net over F₂₅₆, using
  - s-reduction based on digital (1, 4, 65537)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴, 65537, F₂₅₆, 3, 3) (dual of [(65537, 3), 196607, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(256⁴, 65537, F₂₅₆, 2, 3) (dual of [(65537, 2), 131070, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(256⁴, 65537, F₂₅₆, 3) (dual of [65537, 65533, 4]-code or 65537-cap in PG(3,256)), using
        ovoid in PG(3,Â 256) [i]
255. digital (2, 7, 13108)-net over F₂₅₆, using
  - s-reduction based on digital (2, 7, 32640)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁷, 32640, F₂₅₆, 5, 5) (dual of [(32640, 5), 163193, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
256. digital (10, 20, 13108)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256²⁰, 13108, F₂₅₆, 10, 10) (dual of [(13108, 10), 131060, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(256²⁰, 65540, F₂₅₆, 10) (dual of [65540, 65520, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(256²⁰, 65541, F₂₅₆, 10) (dual of [65541, 65521, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
        linear OA(256¹⁹, 65536, F₂₅₆, 10) (dual of [65536, 65517, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        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(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]

(42−10, 42, large)-Net over F₂₅₆ — Digital

Digital (32, 42, large)-net over F₂₅₆, using

t-expansion [i] based on digital (30, 42, large)-net over F₂₅₆, using
- 2 times m-reduction [i] based on digital (30, 44, large)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴⁴, large, F₂₅₆, 14) (dual of [large, large−44, 15]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(42−10, 42, large)-Net in Base 256 — Upper bound on s

There is no (32, 42, large)-net in base 256, because

8 times m-reduction [i] would yield (32, 34, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (42−10, 42, s)-Nets in Base 256

(42−10, 42, 3355648)-Net over F256 — Constructive and digital

(42−10, 42, large)-Net over F256 — Digital

(42−10, 42, large)-Net in Base 256 — Upper bound on s

(42−10, 42, 3355648)-Net over F₂₅₆ — Constructive and digital

(42−10, 42, large)-Net over F₂₅₆ — Digital