Best Known (4, s)-Sequences in Base 9
(4, 29)-Sequence over F9 — Constructive and digital
Digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
(4, 43)-Sequence over F9 — Upper bound on s (digital)
There is no digital (4, 44)-sequence over F9, because
- net from sequence [i] would yield digital (4, m, 45)-net over F9 for arbitrarily large m, but
- m-reduction [i] would yield digital (4, 40, 45)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- construction Y1 [i] would yield
- OA(939, 41, S9, 36), but
- the (dual) Plotkin bound shows that M ≥ 739 044147 071729 616580 416051 031916 488005 / 37 > 939 [i]
- OA(95, 45, S9, 4), but
- discarding factors would yield OA(95, 44, S9, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 60897 > 95 [i]
- discarding factors would yield OA(95, 44, S9, 4), but
- OA(939, 41, S9, 36), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- m-reduction [i] would yield digital (4, 40, 45)-net over F9, but
(4, 47)-Sequence in Base 9 — Upper bound on s
There is no (4, 48)-sequence in base 9, because
- net from sequence [i] would yield (4, m, 49)-net in base 9 for arbitrarily large m, but
- m-reduction [i] would yield (4, 95, 49)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(995, 49, S9, 2, 91), but
- the bound derived from the LP bound by Trinker shows that N ≤ 645 < 93 [i]
- extracting embedded OOA [i] would yield OOA(995, 49, S9, 2, 91), but
- m-reduction [i] would yield (4, 95, 49)-net in base 9, but