Best Known (186−94, 186, s)-Nets in Base 2
(186−94, 186, 53)-Net over F2 — Constructive and digital
Digital (92, 186, 53)-net over F2, using
- t-expansion [i] based on digital (90, 186, 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
(186−94, 186, 60)-Net over F2 — Digital
Digital (92, 186, 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
(186−94, 186, 195)-Net over F2 — Upper bound on s (digital)
There is no digital (92, 186, 196)-net over F2, because
- 2 times m-reduction [i] would yield digital (92, 184, 196)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2184, 196, F2, 92) (dual of [196, 12, 93]-code), but
- residual code [i] would yield linear OA(292, 103, F2, 46) (dual of [103, 11, 47]-code), but
- adding a parity check bit [i] would yield linear OA(293, 104, F2, 47) (dual of [104, 11, 48]-code), but
- residual code [i] would yield linear OA(292, 103, F2, 46) (dual of [103, 11, 47]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2184, 196, F2, 92) (dual of [196, 12, 93]-code), but
(186−94, 186, 222)-Net in Base 2 — Upper bound on s
There is no (92, 186, 223)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 110 193371 294433 586738 089699 322328 070971 836113 829330 112620 > 2186 [i]