Best Known (12, s)-Sequences in Base 5
(12, 32)-Sequence over F5 — Constructive and digital
Digital (12, 32)-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 1 place 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]
(12, 60)-Sequence in Base 5 — Upper bound on s
There is no (12, 61)-sequence in base 5, because
- net from sequence [i] would yield (12, m, 62)-net in base 5 for arbitrarily large m, but
- m-reduction [i] would yield (12, 182, 62)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5182, 62, S5, 3, 170), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3833 616374 949133 159442 297789 188221 688974 434001 615517 846808 198917 588462 406201 023158 493451 474526 647615 448382 566682 994365 692138 671875 / 171 > 5182 [i]
- extracting embedded OOA [i] would yield OOA(5182, 62, S5, 3, 170), but
- m-reduction [i] would yield (12, 182, 62)-net in base 5, but