Best Known (74, 74+75, s)-Nets in Base 2
(74, 74+75, 49)-Net over F2 — Constructive and digital
Digital (74, 149, 49)-net over F2, using
- t-expansion [i] based on digital (70, 149, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [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 (70, 48)-sequence over F2, using
(74, 74+75, 161)-Net over F2 — Upper bound on s (digital)
There is no digital (74, 149, 162)-net over F2, because
- 1 times m-reduction [i] would yield digital (74, 148, 162)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2148, 162, F2, 74) (dual of [162, 14, 75]-code), but
- residual code [i] would yield linear OA(274, 87, F2, 37) (dual of [87, 13, 38]-code), but
- 1 times truncation [i] would yield linear OA(273, 86, F2, 36) (dual of [86, 13, 37]-code), but
- residual code [i] would yield linear OA(274, 87, F2, 37) (dual of [87, 13, 38]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2148, 162, F2, 74) (dual of [162, 14, 75]-code), but
(74, 74+75, 184)-Net in Base 2 — Upper bound on s
There is no (74, 149, 185)-net in base 2, because
- 1 times m-reduction [i] would yield (74, 148, 185)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 375 985048 812874 484612 098970 512021 241608 435812 > 2148 [i]