Best Known (57, s)-Sequences in Base 2
(57, 41)-Sequence over F2 — Constructive and digital
Digital (57, 41)-sequence over F2, using
- t-expansion [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
(57, 65)-Sequence in Base 2 — Upper bound on s
There is no (57, 66)-sequence in base 2, because
- net from sequence [i] would yield (57, m, 67)-net in base 2 for arbitrarily large m, but
- m-reduction [i] would yield (57, 393, 67)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2393, 67, S2, 6, 336), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 9 844827 660206 338998 063191 843261 482770 882003 335161 414642 666888 796811 219058 939207 054525 522909 380525 482349 380843 146099 818496 / 337 > 2393 [i]
- extracting embedded OOA [i] would yield OOA(2393, 67, S2, 6, 336), but
- m-reduction [i] would yield (57, 393, 67)-net in base 2, but