Best Known (178, ∞, s)-Nets in Base 2
(178, ∞, 66)-Net over F2 — Constructive and digital
Digital (178, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (178, 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
(178, ∞, 81)-Net over F2 — Digital
Digital (178, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (178, 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
(178, ∞, 188)-Net in Base 2 — Upper bound on s
There is no (178, m, 189)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (178, 1691, 189)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21691, 189, S2, 9, 1513), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 91 919815 352629 157203 607952 470386 502223 077842 350147 349561 007858 504958 509551 507962 348117 386639 130111 757405 299639 355916 381699 804190 679935 294147 558248 912059 874470 582744 659626 172851 068913 128269 272965 030040 637117 556759 386049 242761 305898 613237 003962 601834 393113 839355 081305 537560 608046 525611 621094 219537 992597 097160 393874 961962 276808 732350 303891 888141 674841 262094 862741 447550 474178 187752 595263 708490 021268 213310 926565 567246 346403 027154 468825 125838 690949 472927 175651 962355 311410 749802 433609 718098 948947 548493 681084 334080 / 757 > 21691 [i]
- extracting embedded OOA [i] would yield OOA(21691, 189, S2, 9, 1513), but