Best Known (73, 164, s)-Nets in Base 2
(73, 164, 49)-Net over F2 — Constructive and digital
Digital (73, 164, 49)-net over F2, using
- t-expansion [i] based on digital (70, 164, 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
(73, 164, 156)-Net over F2 — Upper bound on s (digital)
There is no digital (73, 164, 157)-net over F2, because
- 15 times m-reduction [i] would yield digital (73, 149, 157)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2149, 157, F2, 76) (dual of [157, 8, 77]-code), but
- residual code [i] would yield linear OA(273, 80, F2, 38) (dual of [80, 7, 39]-code), but
- residual code [i] would yield linear OA(235, 41, F2, 19) (dual of [41, 6, 20]-code), but
- 1 times truncation [i] would yield linear OA(234, 40, F2, 18) (dual of [40, 6, 19]-code), but
- residual code [i] would yield linear OA(235, 41, F2, 19) (dual of [41, 6, 20]-code), but
- residual code [i] would yield linear OA(273, 80, F2, 38) (dual of [80, 7, 39]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2149, 157, F2, 76) (dual of [157, 8, 77]-code), but
(73, 164, 158)-Net in Base 2 — Upper bound on s
There is no (73, 164, 159)-net in base 2, because
- 1 times m-reduction [i] would yield (73, 163, 159)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13 941930 552639 337358 641337 015869 624783 120801 571344 > 2163 [i]