Best Known (22, 22+21, s)-Nets in Base 2
(22, 22+21, 21)-Net over F2 — Constructive and digital
Digital (22, 43, 21)-net over F2, using
- t-expansion [i] based on digital (21, 43, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
(22, 22+21, 62)-Net over F2 — Upper bound on s (digital)
There is no digital (22, 43, 63)-net over F2, because
- 1 times m-reduction [i] would yield digital (22, 42, 63)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(242, 63, F2, 20) (dual of [63, 21, 21]-code), but
- residual code [i] would yield linear OA(222, 42, F2, 10) (dual of [42, 20, 11]-code), but
- adding a parity check bit [i] would yield linear OA(223, 43, F2, 11) (dual of [43, 20, 12]-code), but
- residual code [i] would yield linear OA(222, 42, F2, 10) (dual of [42, 20, 11]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(242, 63, F2, 20) (dual of [63, 21, 21]-code), but
(22, 22+21, 67)-Net in Base 2 — Upper bound on s
There is no (22, 43, 68)-net in base 2, because
- 1 times m-reduction [i] would yield (22, 42, 68)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(242, 68, S2, 20), but
- the linear programming bound shows that M ≥ 1 917548 278841 344000 / 388531 > 242 [i]
- extracting embedded orthogonal array [i] would yield OA(242, 68, S2, 20), but