Best Known (1, s)-Sequences in Base 32
(1, 43)-Sequence over F32 — Constructive and digital
Digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
(1, 64)-Sequence over F32 — Upper bound on s (digital)
There is no digital (1, 65)-sequence over F32, because
- net from sequence [i] would yield digital (1, m, 66)-net over F32 for arbitrarily large m, but
- m-reduction [i] would yield digital (1, 63, 66)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3263, 66, F32, 62) (dual of [66, 3, 63]-code), but
- m-reduction [i] would yield digital (1, 63, 66)-net over F32, but
(1, 65)-Sequence in Base 32 — Upper bound on s
There is no (1, 66)-sequence in base 32, because
- net from sequence [i] would yield (1, m, 67)-net in base 32 for arbitrarily large m, but
- m-reduction [i] would yield (1, 65, 67)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(3265, 67, S32, 64), but
- the (dual) Plotkin bound shows that M ≥ 6561 752174 349035 773117 506681 352864 096059 508292 679637 309277 311819 229858 997598 127769 670539 531069 161472 / 65 > 3265 [i]
- extracting embedded orthogonal array [i] would yield OA(3265, 67, S32, 64), but
- m-reduction [i] would yield (1, 65, 67)-net in base 32, but