Best Known (73, 73+89, s)-Nets in Base 2
(73, 73+89, 49)-Net over F2 — Constructive and digital
Digital (73, 162, 49)-net over F2, using
- t-expansion [i] based on digital (70, 162, 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, 73+89, 156)-Net over F2 — Upper bound on s (digital)
There is no digital (73, 162, 157)-net over F2, because
- 13 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, 73+89, 158)-Net in Base 2 — Upper bound on s
There is no (73, 162, 159)-net in base 2, because
- 9 times m-reduction [i] would yield (73, 153, 159)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2153, 159, S2, 80), but
- the (dual) Plotkin bound shows that M ≥ 365375 409332 725729 550921 208179 070754 913983 135744 / 27 > 2153 [i]
- extracting embedded orthogonal array [i] would yield OA(2153, 159, S2, 80), but