Best Known (31, s)-Sequences in Base 4
(31, 33)-Sequence over F4 — Constructive and digital
Digital (31, 33)-sequence over F4, using
- t-expansion [i] based on digital (21, 33)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- T5 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
(31, 42)-Sequence in Base 4 — Constructive
(31, 42)-sequence in base 4, using
- t-expansion [i] based on (30, 42)-sequence in base 4, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
(31, 59)-Sequence over F4 — Digital
Digital (31, 59)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 31 and N(F) ≥ 60, using
(31, 105)-Sequence in Base 4 — Upper bound on s
There is no (31, 106)-sequence in base 4, because
- net from sequence [i] would yield (31, m, 107)-net in base 4 for arbitrarily large m, but
- m-reduction [i] would yield (31, 423, 107)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4423, 107, S4, 4, 392), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 84459 564324 052877 587175 524074 479139 689757 295427 431914 787626 368146 079605 079466 356714 408364 734028 728034 546232 715827 936454 465805 269995 900236 694816 435199 698933 050446 927879 147653 995037 068936 252885 888599 772389 076404 544236 216314 252701 942591 726906 231445 473743 339520 / 131 > 4423 [i]
- extracting embedded OOA [i] would yield OOA(4423, 107, S4, 4, 392), but
- m-reduction [i] would yield (31, 423, 107)-net in base 4, but