Best Known (21, 47, s)-Nets in Base 2
(21, 47, 21)-Net over F2 — Constructive and digital
Digital (21, 47, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
(21, 47, 49)-Net over F2 — Upper bound on s (digital)
There is no digital (21, 47, 50)-net over F2, because
- 2 times m-reduction [i] would yield digital (21, 45, 50)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(245, 50, F2, 24) (dual of [50, 5, 25]-code), but
(21, 47, 50)-Net in Base 2 — Upper bound on s
There is no (21, 47, 51)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(247, 51, S2, 26), but
- adding a parity check bit [i] would yield OA(248, 52, S2, 27), but
- the (dual) Plotkin bound shows that M ≥ 2251 799813 685248 / 7 > 248 [i]
- adding a parity check bit [i] would yield OA(248, 52, S2, 27), but