MinT - Best Known (84−10, 84, s)-Nets in Base 16

Best Known (84−10, 84, s)-Nets in Base 16

(84−10, 84, 6711296)-Net over F₁₆ — Constructive and digital

Digital (74, 84, 6711296)-net over F₁₆, using

trace code for nets [i] based on 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]

(84−10, 84, large)-Net over F₁₆ — Digital

Digital (74, 84, large)-net over F₁₆, using

6 times m-reduction [i] based on digital (74, 90, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁰, large, F₁₆, 16) (dual of [large, large−90, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(84−10, 84, large)-Net in Base 16 — Upper bound on s

There is no (74, 84, large)-net in base 16, because

8 times m-reduction [i] would yield (74, 76, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (84−10, 84, s)-Nets in Base 16

(84−10, 84, 6711296)-Net over F16 — Constructive and digital

(84−10, 84, large)-Net over F16 — Digital

(84−10, 84, large)-Net in Base 16 — Upper bound on s

(84−10, 84, 6711296)-Net over F₁₆ — Constructive and digital

(84−10, 84, large)-Net over F₁₆ — Digital