Best Known (181, ∞, s)-Nets in Base 2
(181, ∞, 66)-Net over F2 — Constructive and digital
Digital (181, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (181, 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
(181, ∞, 82)-Net over F2 — Digital
Digital (181, m, 82)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (181, 81)-sequence over F2, using
- base reduction for sequences [i] based on digital (50, 81)-sequence over F4, using
- s-reduction based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- s-reduction based on digital (50, 90)-sequence over F4, using
- base reduction for sequences [i] based on digital (50, 81)-sequence over F4, using
(181, ∞, 191)-Net in Base 2 — Upper bound on s
There is no (181, m, 192)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (181, 1718, 192)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21718, 192, S2, 9, 1537), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 12292 943258 253202 853825 165503 846279 429612 833967 975200 988965 061600 027064 145360 944275 434553 428006 384168 932816 816731 701364 612328 274374 761463 935038 891988 790950 014335 896121 856694 207791 120447 584097 116322 186085 081728 933788 811240 651424 805734 957319 524103 388539 507581 603008 228031 526909 330084 132523 130452 254980 099742 015112 670832 872900 353124 936510 279298 620709 910305 453782 991649 436476 207808 180506 053072 531698 552242 318544 093943 463462 767651 150251 415746 517996 188245 955301 888844 875031 648703 852253 366693 068301 665346 172925 859061 779616 759808 / 769 > 21718 [i]
- extracting embedded OOA [i] would yield OOA(21718, 192, S2, 9, 1537), but