Best Known (5, s)-Sequences in Base 3
(5, 12)-Sequence over F3 — Constructive and digital
Digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
(5, 16)-Sequence in Base 3 — Upper bound on s
There is no (5, 17)-sequence in base 3, because
- net from sequence [i] would yield (5, m, 18)-net in base 3 for arbitrarily large m, but
- m-reduction [i] would yield (5, 30, 18)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(330, 18, S3, 2, 25), but
- the linear programming bound for OOAs shows that M ≥ 105622 150764 554937 / 455 > 330 [i]
- extracting embedded OOA [i] would yield OOA(330, 18, S3, 2, 25), but
- m-reduction [i] would yield (5, 30, 18)-net in base 3, but