Best Known (145−75, 145, s)-Nets in Base 2
(145−75, 145, 49)-Net over F2 — Constructive and digital
Digital (70, 145, 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]
(145−75, 145, 149)-Net over F2 — Upper bound on s (digital)
There is no digital (70, 145, 150)-net over F2, because
- 3 times m-reduction [i] would yield digital (70, 142, 150)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2142, 150, F2, 72) (dual of [150, 8, 73]-code), but
- residual code [i] would yield linear OA(270, 77, F2, 36) (dual of [77, 7, 37]-code), but
- residual code [i] would yield linear OA(234, 40, F2, 18) (dual of [40, 6, 19]-code), but
- residual code [i] would yield linear OA(270, 77, F2, 36) (dual of [77, 7, 37]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2142, 150, F2, 72) (dual of [150, 8, 73]-code), but
(145−75, 145, 168)-Net in Base 2 — Upper bound on s
There is no (70, 145, 169)-net in base 2, because
- 1 times m-reduction [i] would yield (70, 144, 169)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 030966 262821 418492 316809 121521 526067 724308 > 2144 [i]