Best Known (146−71, 146, s)-Nets in Base 2
(146−71, 146, 50)-Net over F2 — Constructive and digital
Digital (75, 146, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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]
(146−71, 146, 172)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 146, 173)-net over F2, because
- 1 times m-reduction [i] would yield digital (75, 145, 173)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2145, 173, F2, 70) (dual of [173, 28, 71]-code), but
- 3 times code embedding in larger space [i] would yield linear OA(2148, 176, F2, 70) (dual of [176, 28, 71]-code), but
- adding a parity check bit [i] would yield linear OA(2149, 177, F2, 71) (dual of [177, 28, 72]-code), but
- 3 times code embedding in larger space [i] would yield linear OA(2148, 176, F2, 70) (dual of [176, 28, 71]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2145, 173, F2, 70) (dual of [173, 28, 71]-code), but
(146−71, 146, 198)-Net in Base 2 — Upper bound on s
There is no (75, 146, 199)-net in base 2, because
- 1 times m-reduction [i] would yield (75, 145, 199)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 50 054181 304965 337877 616725 078380 224607 717404 > 2145 [i]