Best Known (22, 22+25, s)-Nets in Base 2
(22, 22+25, 21)-Net over F2 — Constructive and digital
Digital (22, 47, 21)-net over F2, using
- t-expansion [i] based on 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
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
(22, 22+25, 52)-Net over F2 — Upper bound on s (digital)
There is no digital (22, 47, 53)-net over F2, because
- 1 times m-reduction [i] would yield digital (22, 46, 53)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- “vT4†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
(22, 22+25, 54)-Net in Base 2 — Upper bound on s
There is no (22, 47, 55)-net in base 2, because
- 1 times m-reduction [i] would yield (22, 46, 55)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(246, 55, S2, 24), but
- the linear programming bound shows that M ≥ 18014 398509 481984 / 221 > 246 [i]
- extracting embedded orthogonal array [i] would yield OA(246, 55, S2, 24), but