Best Known (164, ∞, s)-Nets in Base 2
(164, ∞, 66)-Net over F2 — Constructive and digital
Digital (164, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (164, 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
(164, ∞, 81)-Net over F2 — Digital
Digital (164, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (164, 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
(164, ∞, 174)-Net in Base 2 — Upper bound on s
There is no (164, m, 175)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (164, 1390, 175)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21390, 175, S2, 8, 1226), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 36 423683 632146 431599 092908 269440 765436 536913 734702 027937 621752 624036 075263 631370 189251 511281 617647 499462 136595 077897 507943 271206 213647 619641 189548 324802 937995 044538 611307 272259 854173 086113 528333 076635 879059 972776 904829 423264 105828 458082 158879 047399 243747 351980 833514 936411 869244 700378 570642 324037 990958 429198 725972 245228 977940 618609 964989 717752 973594 810251 387707 762141 585716 968850 223271 502381 634237 960093 875472 078263 549952 / 1227 > 21390 [i]
- extracting embedded OOA [i] would yield OOA(21390, 175, S2, 8, 1226), but