Best Known (13, s)-Sequences in Base 5
(13, 33)-Sequence over F5 — Constructive and digital
Digital (13, 33)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 11, N(F) = 32, and 2 places with degree 2 [i] based on function field F/F5 with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(13, 35)-Sequence over F5 — Digital
Digital (13, 35)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 13 and N(F) ≥ 36, using
(13, 64)-Sequence in Base 5 — Upper bound on s
There is no (13, 65)-sequence in base 5, because
- net from sequence [i] would yield (13, m, 66)-net in base 5 for arbitrarily large m, but
- m-reduction [i] would yield (13, 194, 66)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5194, 66, S5, 3, 181), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 458013 924450 580029 645803 114358 217144 107442 559617 877900 996242 182848 521834 185004 221271 138338 982052 522208 032740 991257 014684 379100 799560 546875 / 91 > 5194 [i]
- extracting embedded OOA [i] would yield OOA(5194, 66, S5, 3, 181), but
- m-reduction [i] would yield (13, 194, 66)-net in base 5, but