Best Known (177−84, 177, s)-Nets in Base 2
(177−84, 177, 53)-Net over F2 — Constructive and digital
Digital (93, 177, 53)-net over F2, using
- t-expansion [i] based on digital (90, 177, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (90, 52)-sequence over F2, using
(177−84, 177, 60)-Net over F2 — Digital
Digital (93, 177, 60)-net over F2, using
- t-expansion [i] based on digital (92, 177, 60)-net over F2, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 92 and N(F) ≥ 60, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
(177−84, 177, 206)-Net over F2 — Upper bound on s (digital)
There is no digital (93, 177, 207)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2177, 207, F2, 84) (dual of [207, 30, 85]-code), but
- 2 times code embedding in larger space [i] would yield linear OA(2179, 209, F2, 84) (dual of [209, 30, 85]-code), but
- adding a parity check bit [i] would yield linear OA(2180, 210, F2, 85) (dual of [210, 30, 86]-code), but
- 2 times code embedding in larger space [i] would yield linear OA(2179, 209, F2, 84) (dual of [209, 30, 85]-code), but
(177−84, 177, 249)-Net in Base 2 — Upper bound on s
There is no (93, 177, 250)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 212724 312941 805803 201477 533689 723274 190956 079120 752336 > 2177 [i]