MinT - Best Known (109, 109+11, s)-Nets in Base 8

Best Known (109, 109+11, s)-Nets in Base 8

(109, 109+11, 6711088)-Net over F₈ — Constructive and digital

Digital (109, 120, 6711088)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 5, 208)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵, 208, F₈, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(8⁵, 208, F₈, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁵, 208, F₈, 3) (dual of [208, 203, 4]-code or 208-cap in PG(4,8)), using
        caps completed using a computer search [i]
2. digital (28, 33, 3355440)-net over F₈, using
  - s-reduction based on digital (28, 33, 4194301)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8³³, 4194301, F₈, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(8³³, large, F₈, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
3. digital (71, 82, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(109, 109+11, 6711296)-Net in Base 8 — Constructive

(109, 120, 6711296)-net in base 8, using

base change [i] based on digital (79, 90, 6711296)-net over F₁₆, using
- trace code for nets [i] based on digital (34, 45, 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, 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]
    247. digital (0, 1, 13108)-net over F₂₅₆ (see above)
    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 (11, 22, 13108)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256²², 13108, F₂₅₆, 11, 11) (dual of [(13108, 11), 144166, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(256²², 65541, F₂₅₆, 11) (dual of [65541, 65519, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(256²², 65542, F₂₅₆, 11) (dual of [65542, 65520, 12]-code), using
        constructionÂ X applied to C([0,5]) âŠ‚ C([0,4]) [i] based on
        linear OA(256²¹, 65537, F₂₅₆, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(256¹⁷, 65537, F₂₅₆, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (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]

(109, 109+11, large)-Net over F₈ — Digital

Digital (109, 120, large)-net over F₈, using

t-expansion [i] based on digital (106, 120, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (106, 124, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁴, large, F₈, 18) (dual of [large, large−124, 19]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²¹, large, F₈, 18) (dual of [large, large−121, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(109, 109+11, large)-Net in Base 8 — Upper bound on s

There is no (109, 120, large)-net in base 8, because

9 times m-reduction [i] would yield (109, 111, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (109, 109+11, s)-Nets in Base 8

(109, 109+11, 6711088)-Net over F8 — Constructive and digital

(109, 109+11, 6711296)-Net in Base 8 — Constructive

(109, 109+11, large)-Net over F8 — Digital

(109, 109+11, large)-Net in Base 8 — Upper bound on s

(109, 109+11, 6711088)-Net over F₈ — Constructive and digital

(109, 109+11, large)-Net over F₈ — Digital