Best Known (8−4, 8, s)-Nets in Base 2
(8−4, 8, 17)-Net over F2 — Constructive and digital
Digital (4, 8, 17)-net over F2, using
(8−4, 8, 17)-Net over F2 — Upper bound on s (digital)
There is no digital (4, 8, 18)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(28, 18, F2, 4) (dual of [18, 10, 5]-code), but
- construction Y1 [i] would yield
- linear OA(27, 12, F2, 4) (dual of [12, 5, 5]-code), but
- OA(210, 18, S2, 6), but
- discarding factors would yield OA(210, 16, S2, 6), but
- the linear programming bound shows that M ≥ 27136 / 21 > 210 [i]
- discarding factors would yield OA(210, 16, S2, 6), but
- construction Y1 [i] would yield
(8−4, 8, 20)-Net in Base 2 — Upper bound on s
There is no (4, 8, 21)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 274 > 28 [i]
- extracting embedded orthogonal array [i] would yield OA(28, 21, S2, 4), but
- the linear programming bound shows that M ≥ 8960 / 33 > 28 [i]