Best Known (67, s)-Sequences in Base 2
(67, 42)-Sequence over F2 — Constructive and digital
Digital (67, 42)-sequence over F2, using
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
(67, 47)-Sequence over F2 — Digital
Digital (67, 47)-sequence over F2, using
- t-expansion [i] based on digital (65, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 65 and N(F) ≥ 48, using
(67, 75)-Sequence in Base 2 — Upper bound on s
There is no (67, 76)-sequence in base 2, because
- net from sequence [i] would yield (67, m, 77)-net in base 2 for arbitrarily large m, but
- m-reduction [i] would yield (67, 453, 77)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2453, 77, S2, 6, 386), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 163202 128134 481187 670657 211928 595733 507047 201711 208198 345045 346834 142181 869507 762315 583452 706299 037320 438494 030470 406366 742586 287730 982912 / 129 > 2453 [i]
- extracting embedded OOA [i] would yield OOA(2453, 77, S2, 6, 386), but
- m-reduction [i] would yield (67, 453, 77)-net in base 2, but