Best Known (74, 74+71, s)-Nets in Base 2
(74, 74+71, 49)-Net over F2 — Constructive and digital
Digital (74, 145, 49)-net over F2, using
- t-expansion [i] based on 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]
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
(74, 74+71, 171)-Net over F2 — Upper bound on s (digital)
There is no digital (74, 145, 172)-net over F2, because
- 1 times m-reduction [i] would yield digital (74, 144, 172)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2144, 172, F2, 70) (dual of [172, 28, 71]-code), but
- 4 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
- 4 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(2144, 172, F2, 70) (dual of [172, 28, 71]-code), but
(74, 74+71, 193)-Net in Base 2 — Upper bound on s
There is no (74, 145, 194)-net in base 2, because
- 1 times m-reduction [i] would yield (74, 144, 194)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 24 131703 366748 957754 995562 713595 329328 933256 > 2144 [i]