Best Known (18, s)-Sequences in Base 5
(18, 42)-Sequence over F5 — Constructive and digital
Digital (18, 42)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 17, N(F) = 42, and 1 place with degree 2 [i] based on function field F/F5 with g(F) = 17 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
(18, 85)-Sequence in Base 5 — Upper bound on s
There is no (18, 86)-sequence in base 5, because
- net from sequence [i] would yield (18, m, 87)-net in base 5 for arbitrarily large m, but
- m-reduction [i] would yield (18, 257, 87)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5257, 87, S5, 3, 239), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 295425 283264 166693 807952 779420 059935 667400 054665 442207 753555 520525 288770 474364 058695 726125 520843 233984 082094 091922 449539 549264 842024 792004 047950 566558 711216 202937 066555 023193 359375 / 2 > 5257 [i]
- extracting embedded OOA [i] would yield OOA(5257, 87, S5, 3, 239), but
- m-reduction [i] would yield (18, 257, 87)-net in base 5, but