Best Known (24, 24+28, s)-Nets in Base 2
(24, 24+28, 21)-Net over F2 — Constructive and digital
Digital (24, 52, 21)-net over F2, using
- t-expansion [i] based on digital (21, 52, 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
(24, 24+28, 22)-Net over F2 — Digital
Digital (24, 52, 22)-net over F2, using
- t-expansion [i] based on digital (23, 52, 22)-net over F2, using
- net from sequence [i] based on digital (23, 21)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 23 and N(F) ≥ 22, using
- net from sequence [i] based on digital (23, 21)-sequence over F2, using
(24, 24+28, 55)-Net over F2 — Upper bound on s (digital)
There is no digital (24, 52, 56)-net over F2, because
- 4 times m-reduction [i] would yield digital (24, 48, 56)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
(24, 24+28, 56)-Net in Base 2 — Upper bound on s
There is no (24, 52, 57)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(252, 57, S2, 28), but
- adding a parity check bit [i] would yield OA(253, 58, S2, 29), but
- the (dual) Plotkin bound shows that M ≥ 144115 188075 855872 / 15 > 253 [i]
- adding a parity check bit [i] would yield OA(253, 58, S2, 29), but