Best Known (169, 169+∞, s)-Nets in Base 2
(169, 169+∞, 66)-Net over F2 — Constructive and digital
Digital (169, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (169, 65)-sequence over F2, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
(169, 169+∞, 81)-Net over F2 — Digital
Digital (169, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (169, 80)-sequence over F2, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
(169, 169+∞, 179)-Net in Base 2 — Upper bound on s
There is no (169, m, 180)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (169, 1430, 180)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21430, 180, S2, 8, 1261), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 19 727037 895776 188686 585189 056144 957731 715503 877237 638023 441459 214722 326824 054168 677014 226343 682576 537283 414605 620388 451713 395852 499979 161848 941355 696449 081671 672197 822085 456041 449909 603696 309801 780239 696473 352244 954615 427604 303332 082428 604815 764923 250732 440334 851499 731526 144506 460610 961422 318708 830626 903753 513691 475867 968022 449568 078417 547844 670674 127096 491628 943190 883481 151566 529941 486180 586165 608095 906114 279287 806875 049851 355136 / 631 > 21430 [i]
- extracting embedded OOA [i] would yield OOA(21430, 180, S2, 8, 1261), but