MinT - Best Known (51, 51+16, s)-Nets in Base 256

Best Known (51, 51+16, s)-Nets in Base 256

(51, 51+16, 1114368)-Net over F₂₅₆ — Constructive and digital

Digital (51, 67, 1114368)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 65793)-net over F₂₅₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 257)-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, 257)-net over F₂₅₆ (see above)
    3. digital (0, 0, 257)-net over F₂₅₆ (see above)
    4. digital (0, 0, 257)-net over F₂₅₆ (see above)
    5. digital (0, 0, 257)-net over F₂₅₆ (see above)
    6. digital (0, 0, 257)-net over F₂₅₆ (see above)
    7. digital (0, 0, 257)-net over F₂₅₆ (see above)
    8. digital (0, 0, 257)-net over F₂₅₆ (see above)
    9. digital (0, 0, 257)-net over F₂₅₆ (see above)
    10. digital (0, 0, 257)-net over F₂₅₆ (see above)
    11. digital (0, 0, 257)-net over F₂₅₆ (see above)
    12. digital (0, 0, 257)-net over F₂₅₆ (see above)
    13. digital (0, 0, 257)-net over F₂₅₆ (see above)
    14. digital (0, 0, 257)-net over F₂₅₆ (see above)
    15. digital (0, 0, 257)-net over F₂₅₆ (see above)
    16. digital (0, 0, 257)-net over F₂₅₆ (see above)
    17. digital (0, 0, 257)-net over F₂₅₆ (see above)
    18. digital (0, 0, 257)-net over F₂₅₆ (see above)
    19. digital (0, 0, 257)-net over F₂₅₆ (see above)
    20. digital (0, 0, 257)-net over F₂₅₆ (see above)
    21. digital (0, 0, 257)-net over F₂₅₆ (see above)
    22. digital (0, 0, 257)-net over F₂₅₆ (see above)
    23. digital (0, 0, 257)-net over F₂₅₆ (see above)
    24. digital (0, 0, 257)-net over F₂₅₆ (see above)
    25. digital (0, 0, 257)-net over F₂₅₆ (see above)
    26. digital (0, 0, 257)-net over F₂₅₆ (see above)
    27. digital (0, 0, 257)-net over F₂₅₆ (see above)
    28. digital (0, 0, 257)-net over F₂₅₆ (see above)
    29. digital (0, 0, 257)-net over F₂₅₆ (see above)
    30. digital (0, 0, 257)-net over F₂₅₆ (see above)
    31. digital (0, 0, 257)-net over F₂₅₆ (see above)
    32. digital (0, 0, 257)-net over F₂₅₆ (see above)
    33. digital (0, 0, 257)-net over F₂₅₆ (see above)
    34. digital (0, 0, 257)-net over F₂₅₆ (see above)
    35. digital (0, 0, 257)-net over F₂₅₆ (see above)
    36. digital (0, 0, 257)-net over F₂₅₆ (see above)
    37. digital (0, 0, 257)-net over F₂₅₆ (see above)
    38. digital (0, 0, 257)-net over F₂₅₆ (see above)
    39. digital (0, 0, 257)-net over F₂₅₆ (see above)
    40. digital (0, 0, 257)-net over F₂₅₆ (see above)
    41. digital (0, 0, 257)-net over F₂₅₆ (see above)
    42. digital (0, 0, 257)-net over F₂₅₆ (see above)
    43. digital (0, 0, 257)-net over F₂₅₆ (see above)
    44. digital (0, 0, 257)-net over F₂₅₆ (see above)
    45. digital (0, 0, 257)-net over F₂₅₆ (see above)
    46. digital (0, 0, 257)-net over F₂₅₆ (see above)
    47. digital (0, 0, 257)-net over F₂₅₆ (see above)
    48. digital (0, 0, 257)-net over F₂₅₆ (see above)
    49. digital (0, 0, 257)-net over F₂₅₆ (see above)
    50. digital (0, 0, 257)-net over F₂₅₆ (see above)
    51. digital (0, 0, 257)-net over F₂₅₆ (see above)
    52. digital (0, 0, 257)-net over F₂₅₆ (see above)
    53. digital (0, 0, 257)-net over F₂₅₆ (see above)
    54. digital (0, 0, 257)-net over F₂₅₆ (see above)
    55. digital (0, 0, 257)-net over F₂₅₆ (see above)
    56. digital (0, 0, 257)-net over F₂₅₆ (see above)
    57. digital (0, 0, 257)-net over F₂₅₆ (see above)
    58. digital (0, 0, 257)-net over F₂₅₆ (see above)
    59. digital (0, 0, 257)-net over F₂₅₆ (see above)
    60. digital (0, 0, 257)-net over F₂₅₆ (see above)
    61. digital (0, 0, 257)-net over F₂₅₆ (see above)
    62. digital (0, 0, 257)-net over F₂₅₆ (see above)
    63. digital (0, 0, 257)-net over F₂₅₆ (see above)
    64. digital (0, 0, 257)-net over F₂₅₆ (see above)
    65. digital (0, 0, 257)-net over F₂₅₆ (see above)
    66. digital (0, 0, 257)-net over F₂₅₆ (see above)
    67. digital (0, 0, 257)-net over F₂₅₆ (see above)
    68. digital (0, 0, 257)-net over F₂₅₆ (see above)
    69. digital (0, 0, 257)-net over F₂₅₆ (see above)
    70. digital (0, 0, 257)-net over F₂₅₆ (see above)
    71. digital (0, 0, 257)-net over F₂₅₆ (see above)
    72. digital (0, 0, 257)-net over F₂₅₆ (see above)
    73. digital (0, 0, 257)-net over F₂₅₆ (see above)
    74. digital (0, 0, 257)-net over F₂₅₆ (see above)
    75. digital (0, 0, 257)-net over F₂₅₆ (see above)
    76. digital (0, 0, 257)-net over F₂₅₆ (see above)
    77. digital (0, 0, 257)-net over F₂₅₆ (see above)
    78. digital (0, 0, 257)-net over F₂₅₆ (see above)
    79. digital (0, 0, 257)-net over F₂₅₆ (see above)
    80. digital (0, 0, 257)-net over F₂₅₆ (see above)
    81. digital (0, 0, 257)-net over F₂₅₆ (see above)
    82. digital (0, 0, 257)-net over F₂₅₆ (see above)
    83. digital (0, 0, 257)-net over F₂₅₆ (see above)
    84. digital (0, 0, 257)-net over F₂₅₆ (see above)
    85. digital (0, 0, 257)-net over F₂₅₆ (see above)
    86. digital (0, 0, 257)-net over F₂₅₆ (see above)
    87. digital (0, 0, 257)-net over F₂₅₆ (see above)
    88. digital (0, 0, 257)-net over F₂₅₆ (see above)
    89. digital (0, 0, 257)-net over F₂₅₆ (see above)
    90. digital (0, 0, 257)-net over F₂₅₆ (see above)
    91. digital (0, 0, 257)-net over F₂₅₆ (see above)
    92. digital (0, 0, 257)-net over F₂₅₆ (see above)
    93. digital (0, 0, 257)-net over F₂₅₆ (see above)
    94. digital (0, 0, 257)-net over F₂₅₆ (see above)
    95. digital (0, 0, 257)-net over F₂₅₆ (see above)
    96. digital (0, 0, 257)-net over F₂₅₆ (see above)
    97. digital (0, 0, 257)-net over F₂₅₆ (see above)
    98. digital (0, 0, 257)-net over F₂₅₆ (see above)
    99. digital (0, 0, 257)-net over F₂₅₆ (see above)
    100. digital (0, 0, 257)-net over F₂₅₆ (see above)
    101. digital (0, 0, 257)-net over F₂₅₆ (see above)
    102. digital (0, 0, 257)-net over F₂₅₆ (see above)
    103. digital (0, 0, 257)-net over F₂₅₆ (see above)
    104. digital (0, 0, 257)-net over F₂₅₆ (see above)
    105. digital (0, 0, 257)-net over F₂₅₆ (see above)
    106. digital (0, 0, 257)-net over F₂₅₆ (see above)
    107. digital (0, 0, 257)-net over F₂₅₆ (see above)
    108. digital (0, 0, 257)-net over F₂₅₆ (see above)
    109. digital (0, 0, 257)-net over F₂₅₆ (see above)
    110. digital (0, 0, 257)-net over F₂₅₆ (see above)
    111. digital (0, 0, 257)-net over F₂₅₆ (see above)
    112. digital (0, 0, 257)-net over F₂₅₆ (see above)
    113. digital (0, 0, 257)-net over F₂₅₆ (see above)
    114. digital (0, 0, 257)-net over F₂₅₆ (see above)
    115. digital (0, 0, 257)-net over F₂₅₆ (see above)
    116. digital (0, 0, 257)-net over F₂₅₆ (see above)
    117. digital (0, 0, 257)-net over F₂₅₆ (see above)
    118. digital (0, 0, 257)-net over F₂₅₆ (see above)
    119. digital (0, 0, 257)-net over F₂₅₆ (see above)
    120. digital (0, 0, 257)-net over F₂₅₆ (see above)
    121. digital (0, 0, 257)-net over F₂₅₆ (see above)
    122. digital (0, 0, 257)-net over F₂₅₆ (see above)
    123. digital (0, 0, 257)-net over F₂₅₆ (see above)
    124. digital (0, 0, 257)-net over F₂₅₆ (see above)
    125. digital (0, 0, 257)-net over F₂₅₆ (see above)
    126. digital (0, 0, 257)-net over F₂₅₆ (see above)
    127. digital (0, 0, 257)-net over F₂₅₆ (see above)
    128. digital (0, 0, 257)-net over F₂₅₆ (see above)
    129. digital (0, 0, 257)-net over F₂₅₆ (see above)
    130. digital (0, 0, 257)-net over F₂₅₆ (see above)
    131. digital (0, 0, 257)-net over F₂₅₆ (see above)
    132. digital (0, 0, 257)-net over F₂₅₆ (see above)
    133. digital (0, 0, 257)-net over F₂₅₆ (see above)
    134. digital (0, 0, 257)-net over F₂₅₆ (see above)
    135. digital (0, 0, 257)-net over F₂₅₆ (see above)
    136. digital (0, 0, 257)-net over F₂₅₆ (see above)
    137. digital (0, 0, 257)-net over F₂₅₆ (see above)
    138. digital (0, 0, 257)-net over F₂₅₆ (see above)
    139. digital (0, 0, 257)-net over F₂₅₆ (see above)
    140. digital (0, 0, 257)-net over F₂₅₆ (see above)
    141. digital (0, 0, 257)-net over F₂₅₆ (see above)
    142. digital (0, 0, 257)-net over F₂₅₆ (see above)
    143. digital (0, 0, 257)-net over F₂₅₆ (see above)
    144. digital (0, 0, 257)-net over F₂₅₆ (see above)
    145. digital (0, 0, 257)-net over F₂₅₆ (see above)
    146. digital (0, 0, 257)-net over F₂₅₆ (see above)
    147. digital (0, 0, 257)-net over F₂₅₆ (see above)
    148. digital (0, 0, 257)-net over F₂₅₆ (see above)
    149. digital (0, 0, 257)-net over F₂₅₆ (see above)
    150. digital (0, 0, 257)-net over F₂₅₆ (see above)
    151. digital (0, 0, 257)-net over F₂₅₆ (see above)
    152. digital (0, 0, 257)-net over F₂₅₆ (see above)
    153. digital (0, 0, 257)-net over F₂₅₆ (see above)
    154. digital (0, 0, 257)-net over F₂₅₆ (see above)
    155. digital (0, 0, 257)-net over F₂₅₆ (see above)
    156. digital (0, 0, 257)-net over F₂₅₆ (see above)
    157. digital (0, 0, 257)-net over F₂₅₆ (see above)
    158. digital (0, 0, 257)-net over F₂₅₆ (see above)
    159. digital (0, 0, 257)-net over F₂₅₆ (see above)
    160. digital (0, 0, 257)-net over F₂₅₆ (see above)
    161. digital (0, 0, 257)-net over F₂₅₆ (see above)
    162. digital (0, 0, 257)-net over F₂₅₆ (see above)
    163. digital (0, 0, 257)-net over F₂₅₆ (see above)
    164. digital (0, 0, 257)-net over F₂₅₆ (see above)
    165. digital (0, 0, 257)-net over F₂₅₆ (see above)
    166. digital (0, 0, 257)-net over F₂₅₆ (see above)
    167. digital (0, 0, 257)-net over F₂₅₆ (see above)
    168. digital (0, 0, 257)-net over F₂₅₆ (see above)
    169. digital (0, 0, 257)-net over F₂₅₆ (see above)
    170. digital (0, 0, 257)-net over F₂₅₆ (see above)
    171. digital (0, 0, 257)-net over F₂₅₆ (see above)
    172. digital (0, 0, 257)-net over F₂₅₆ (see above)
    173. digital (0, 0, 257)-net over F₂₅₆ (see above)
    174. digital (0, 0, 257)-net over F₂₅₆ (see above)
    175. digital (0, 0, 257)-net over F₂₅₆ (see above)
    176. digital (0, 0, 257)-net over F₂₅₆ (see above)
    177. digital (0, 0, 257)-net over F₂₅₆ (see above)
    178. digital (0, 0, 257)-net over F₂₅₆ (see above)
    179. digital (0, 0, 257)-net over F₂₅₆ (see above)
    180. digital (0, 0, 257)-net over F₂₅₆ (see above)
    181. digital (0, 0, 257)-net over F₂₅₆ (see above)
    182. digital (0, 0, 257)-net over F₂₅₆ (see above)
    183. digital (0, 0, 257)-net over F₂₅₆ (see above)
    184. digital (0, 0, 257)-net over F₂₅₆ (see above)
    185. digital (0, 0, 257)-net over F₂₅₆ (see above)
    186. digital (0, 0, 257)-net over F₂₅₆ (see above)
    187. digital (0, 0, 257)-net over F₂₅₆ (see above)
    188. digital (0, 0, 257)-net over F₂₅₆ (see above)
    189. digital (0, 0, 257)-net over F₂₅₆ (see above)
    190. digital (0, 0, 257)-net over F₂₅₆ (see above)
    191. digital (0, 0, 257)-net over F₂₅₆ (see above)
    192. digital (0, 0, 257)-net over F₂₅₆ (see above)
    193. digital (0, 0, 257)-net over F₂₅₆ (see above)
    194. digital (0, 0, 257)-net over F₂₅₆ (see above)
    195. digital (0, 0, 257)-net over F₂₅₆ (see above)
    196. digital (0, 0, 257)-net over F₂₅₆ (see above)
    197. digital (0, 0, 257)-net over F₂₅₆ (see above)
    198. digital (0, 0, 257)-net over F₂₅₆ (see above)
    199. digital (0, 0, 257)-net over F₂₅₆ (see above)
    200. digital (0, 0, 257)-net over F₂₅₆ (see above)
    201. digital (0, 0, 257)-net over F₂₅₆ (see above)
    202. digital (0, 0, 257)-net over F₂₅₆ (see above)
    203. digital (0, 0, 257)-net over F₂₅₆ (see above)
    204. digital (0, 0, 257)-net over F₂₅₆ (see above)
    205. digital (0, 0, 257)-net over F₂₅₆ (see above)
    206. digital (0, 0, 257)-net over F₂₅₆ (see above)
    207. digital (0, 0, 257)-net over F₂₅₆ (see above)
    208. digital (0, 0, 257)-net over F₂₅₆ (see above)
    209. digital (0, 0, 257)-net over F₂₅₆ (see above)
    210. digital (0, 0, 257)-net over F₂₅₆ (see above)
    211. digital (0, 0, 257)-net over F₂₅₆ (see above)
    212. digital (0, 0, 257)-net over F₂₅₆ (see above)
    213. digital (0, 0, 257)-net over F₂₅₆ (see above)
    214. digital (0, 0, 257)-net over F₂₅₆ (see above)
    215. digital (0, 0, 257)-net over F₂₅₆ (see above)
    216. digital (0, 0, 257)-net over F₂₅₆ (see above)
    217. digital (0, 0, 257)-net over F₂₅₆ (see above)
    218. digital (0, 0, 257)-net over F₂₅₆ (see above)
    219. digital (0, 0, 257)-net over F₂₅₆ (see above)
    220. digital (0, 0, 257)-net over F₂₅₆ (see above)
    221. digital (0, 0, 257)-net over F₂₅₆ (see above)
    222. digital (0, 0, 257)-net over F₂₅₆ (see above)
    223. digital (0, 0, 257)-net over F₂₅₆ (see above)
    224. digital (0, 0, 257)-net over F₂₅₆ (see above)
    225. digital (0, 0, 257)-net over F₂₅₆ (see above)
    226. digital (0, 0, 257)-net over F₂₅₆ (see above)
    227. digital (0, 0, 257)-net over F₂₅₆ (see above)
    228. digital (0, 0, 257)-net over F₂₅₆ (see above)
    229. digital (0, 0, 257)-net over F₂₅₆ (see above)
    230. digital (0, 0, 257)-net over F₂₅₆ (see above)
    231. digital (0, 0, 257)-net over F₂₅₆ (see above)
    232. digital (0, 0, 257)-net over F₂₅₆ (see above)
    233. digital (0, 0, 257)-net over F₂₅₆ (see above)
    234. digital (0, 0, 257)-net over F₂₅₆ (see above)
    235. digital (0, 0, 257)-net over F₂₅₆ (see above)
    236. digital (0, 0, 257)-net over F₂₅₆ (see above)
    237. digital (0, 0, 257)-net over F₂₅₆ (see above)
    238. digital (0, 0, 257)-net over F₂₅₆ (see above)
    239. digital (0, 0, 257)-net over F₂₅₆ (see above)
    240. digital (0, 0, 257)-net over F₂₅₆ (see above)
    241. digital (0, 0, 257)-net over F₂₅₆ (see above)
    242. digital (0, 0, 257)-net over F₂₅₆ (see above)
    243. digital (0, 0, 257)-net over F₂₅₆ (see above)
    244. digital (0, 0, 257)-net over F₂₅₆ (see above)
    245. digital (0, 0, 257)-net over F₂₅₆ (see above)
    246. digital (0, 0, 257)-net over F₂₅₆ (see above)
    247. digital (0, 0, 257)-net over F₂₅₆ (see above)
    248. digital (0, 0, 257)-net over F₂₅₆ (see above)
    249. digital (0, 1, 257)-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]
    250. digital (0, 1, 257)-net over F₂₅₆ (see above)
    251. digital (0, 1, 257)-net over F₂₅₆ (see above)
    252. digital (0, 1, 257)-net over F₂₅₆ (see above)
    253. digital (0, 2, 257)-net over F₂₅₆, using
      - kÂ =Â 2 construction [i]
    254. digital (0, 2, 257)-net over F₂₅₆ (see above)
    255. digital (0, 4, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    256. digital (1, 9, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (30, 46, 1048575)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴⁶, 1048575, F₂₅₆, 16, 16) (dual of [(1048575, 16), 16777154, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(256⁴⁶, 8388600, F₂₅₆, 16) (dual of [8388600, 8388554, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴⁶, large, F₂₅₆, 16) (dual of [large, large−46, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(51, 51+16, large)-Net over F₂₅₆ — Digital

Digital (51, 67, large)-net over F₂₅₆, using

t-expansion [i] based on digital (47, 67, large)-net over F₂₅₆, using
- 1 times m-reduction [i] based on digital (47, 68, large)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁶⁸, large, F₂₅₆, 21) (dual of [large, large−68, 22]-code), using
    - 7 times code embedding in larger space [i] based on linear OA(256⁶¹, large, F₂₅₆, 21) (dual of [large, large−61, 22]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(51, 51+16, large)-Net in Base 256 — Upper bound on s

There is no (51, 67, large)-net in base 256, because

14 times m-reduction [i] would yield (51, 53, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (51, 51+16, s)-Nets in Base 256

(51, 51+16, 1114368)-Net over F256 — Constructive and digital

(51, 51+16, large)-Net over F256 — Digital

(51, 51+16, large)-Net in Base 256 — Upper bound on s

(51, 51+16, 1114368)-Net over F₂₅₆ — Constructive and digital

(51, 51+16, large)-Net over F₂₅₆ — Digital