Best Known (48, s)-Sequences in Base 2
(48, 34)-Sequence over F2 — Constructive and digital
Digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(48, 35)-Sequence over F2 — Digital
Digital (48, 35)-sequence over F2, using
- t-expansion [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
(48, 55)-Sequence in Base 2 — Upper bound on s
There is no (48, 56)-sequence in base 2, because
- net from sequence [i] would yield (48, m, 57)-net in base 2 for arbitrarily large m, but
- m-reduction [i] would yield (48, 391, 57)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2391, 57, S2, 7, 343), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 250911 975458 640043 623792 927357 714532 710747 779674 323964 381494 732398 236756 240894 237173 486047 511107 142800 656607 042273 804288 / 43 > 2391 [i]
- extracting embedded OOA [i] would yield OOA(2391, 57, S2, 7, 343), but
- m-reduction [i] would yield (48, 391, 57)-net in base 2, but