Best Known (69, 69+75, s)-Nets in Base 2
(69, 69+75, 48)-Net over F2 — Constructive and digital
Digital (69, 144, 48)-net over F2, using
- net from sequence [i] based on digital (69, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using
(69, 69+75, 49)-Net over F2 — Digital
Digital (69, 144, 49)-net over F2, using
- t-expansion [i] based on digital (68, 144, 49)-net over F2, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 68 and N(F) ≥ 49, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
(69, 69+75, 147)-Net over F2 — Upper bound on s (digital)
There is no digital (69, 144, 148)-net over F2, because
- 3 times m-reduction [i] would yield digital (69, 141, 148)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2141, 148, F2, 72) (dual of [148, 7, 73]-code), but
(69, 69+75, 149)-Net in Base 2 — Upper bound on s
There is no (69, 144, 150)-net in base 2, because
- 5 times m-reduction [i] would yield (69, 139, 150)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2139, 150, S2, 70), but
- the linear programming bound shows that M ≥ 6082 528252 899227 461853 867158 864840 600756 158464 / 8127 > 2139 [i]
- extracting embedded orthogonal array [i] would yield OA(2139, 150, S2, 70), but