Best Known (58−30, 58, s)-Nets in Base 2
(58−30, 58, 21)-Net over F2 — Constructive and digital
Digital (28, 58, 21)-net over F2, using
- t-expansion [i] based on digital (21, 58, 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
(58−30, 58, 25)-Net over F2 — Digital
Digital (28, 58, 25)-net over F2, using
- net from sequence [i] based on digital (28, 24)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 28 and N(F) ≥ 25, using
(58−30, 58, 65)-Net over F2 — Upper bound on s (digital)
There is no digital (28, 58, 66)-net over F2, because
- 2 times m-reduction [i] would yield digital (28, 56, 66)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
- adding a parity check bit [i] would yield linear OA(257, 67, F2, 29) (dual of [67, 10, 30]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
(58−30, 58, 66)-Net in Base 2 — Upper bound on s
There is no (28, 58, 67)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(258, 67, S2, 30), but
- the linear programming bound shows that M ≥ 10 376293 541461 622784 / 35 > 258 [i]