Best Known (21, 21+21, s)-Nets in Base 2
(21, 21+21, 21)-Net over F2 — Constructive and digital
Digital (21, 42, 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
(21, 21+21, 53)-Net over F2 — Upper bound on s (digital)
There is no digital (21, 42, 54)-net over F2, because
- 1 times m-reduction [i] would yield digital (21, 41, 54)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(241, 54, F2, 20) (dual of [54, 13, 21]-code), but
- adding a parity check bit [i] would yield linear OA(242, 55, F2, 21) (dual of [55, 13, 22]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(241, 54, F2, 20) (dual of [54, 13, 21]-code), but
(21, 21+21, 61)-Net in Base 2 — Upper bound on s
There is no (21, 42, 62)-net in base 2, because
- 1 times m-reduction [i] would yield (21, 41, 62)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(241, 62, S2, 20), but
- the linear programming bound shows that M ≥ 249180 121179 619328 / 104975 > 241 [i]
- extracting embedded orthogonal array [i] would yield OA(241, 62, S2, 20), but