Best Known (51−29, 51, s)-Nets in Base 2
(51−29, 51, 21)-Net over F2 — Constructive and digital
Digital (22, 51, 21)-net over F2, using
- t-expansion [i] based on digital (21, 51, 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
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
(51−29, 51, 52)-Net in Base 2 — Upper bound on s
There is no (22, 51, 53)-net in base 2, because
- 3 times m-reduction [i] would yield (22, 48, 53)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(248, 53, S2, 26), but
- adding a parity check bit [i] would yield OA(249, 54, S2, 27), but
- the (dual) Plotkin bound shows that M ≥ 4503 599627 370496 / 7 > 249 [i]
- adding a parity check bit [i] would yield OA(249, 54, S2, 27), but
- extracting embedded orthogonal array [i] would yield OA(248, 53, S2, 26), but