Best Known (168, ∞, s)-Nets in Base 2
(168, ∞, 66)-Net over F2 — Constructive and digital
Digital (168, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (168, 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
(168, ∞, 81)-Net over F2 — Digital
Digital (168, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (168, 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
(168, ∞, 178)-Net in Base 2 — Upper bound on s
There is no (168, m, 179)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (168, 1422, 179)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21422, 179, S2, 8, 1254), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 154581 692848 585062 891398 436457 461194 575048 540830 198923 705901 871199 701968 006636 816863 839828 982640 671755 350552 138356 300697 019974 088914 087464 601314 746021 214801 314637 657067 828898 291670 272449 898135 447873 207335 411010 390857 655718 002471 046297 268575 951847 690194 227753 941678 026416 561702 763822 087964 835093 421512 583747 880272 478855 987230 597602 272451 831782 574250 152586 670079 829585 903713 272389 675604 068968 800244 567665 096402 862740 089412 871602 044928 / 1255 > 21422 [i]
- extracting embedded OOA [i] would yield OOA(21422, 179, S2, 8, 1254), but