Best Known (178−86, 178, s)-Nets in Base 2
(178−86, 178, 53)-Net over F2 — Constructive and digital
Digital (92, 178, 53)-net over F2, using
- t-expansion [i] based on digital (90, 178, 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
(178−86, 178, 60)-Net over F2 — Digital
Digital (92, 178, 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
(178−86, 178, 205)-Net over F2 — Upper bound on s (digital)
There is no digital (92, 178, 206)-net over F2, because
- 2 times m-reduction [i] would yield digital (92, 176, 206)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 206, F2, 84) (dual of [206, 30, 85]-code), but
- 3 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
- 3 times code embedding in larger space [i] would yield linear OA(2179, 209, F2, 84) (dual of [209, 30, 85]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 206, F2, 84) (dual of [206, 30, 85]-code), but
(178−86, 178, 239)-Net in Base 2 — Upper bound on s
There is no (92, 178, 240)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 430199 052902 883167 558155 532044 110005 605796 953853 582873 > 2178 [i]