Best Known (84, 84+59, s)-Nets in Base 2
(84, 84+59, 56)-Net over F2 — Constructive and digital
Digital (84, 143, 56)-net over F2, using
- 1 times m-reduction [i] based on digital (84, 144, 56)-net over F2, using
- trace code for nets [i] based on digital (12, 72, 28)-net over F4, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 12 and N(F) ≥ 28, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- trace code for nets [i] based on digital (12, 72, 28)-net over F4, using
(84, 84+59, 65)-Net over F2 — Digital
Digital (84, 143, 65)-net over F2, using
(84, 84+59, 297)-Net in Base 2 — Upper bound on s
There is no (84, 143, 298)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 142, 298)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2142, 298, S2, 58), but
- 1 times code embedding in larger space [i] would yield OA(2143, 299, S2, 58), but
- adding a parity check bit [i] would yield OA(2144, 300, S2, 59), but
- the linear programming bound shows that M ≥ 13 678088 242923 667556 600263 770877 103665 057371 087380 936782 241764 391649 337574 853234 670350 342175 183220 002167 090964 284963 769143 998528 572887 401408 166266 488986 476908 076314 671470 844351 653198 056605 676746 079987 245754 524021 018543 185107 747357 465099 423571 391220 285440 / 511695 224298 586525 211471 124783 467189 485286 772501 732197 706363 184358 810101 284218 553132 471318 324678 593874 150903 851104 221947 201297 793196 366515 952543 045944 571691 389329 298134 869463 666150 935271 854871 838445 853309 987223 > 2144 [i]
- adding a parity check bit [i] would yield OA(2144, 300, S2, 59), but
- 1 times code embedding in larger space [i] would yield OA(2143, 299, S2, 58), but
- extracting embedded orthogonal array [i] would yield OA(2142, 298, S2, 58), but