Best Known (51−28, 51, s)-Nets in Base 2
(51−28, 51, 21)-Net over F2 — Constructive and digital
Digital (23, 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
(51−28, 51, 22)-Net over F2 — Digital
Digital (23, 51, 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
(51−28, 51, 53)-Net over F2 — Upper bound on s (digital)
There is no digital (23, 51, 54)-net over F2, because
- 4 times m-reduction [i] would yield digital (23, 47, 54)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
(51−28, 51, 54)-Net in Base 2 — Upper bound on s
There is no (23, 51, 55)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(251, 55, S2, 28), but
- adding a parity check bit [i] would yield OA(252, 56, S2, 29), but
- the (dual) Plotkin bound shows that M ≥ 72057 594037 927936 / 15 > 252 [i]
- adding a parity check bit [i] would yield OA(252, 56, S2, 29), but