Best Known (157−82, 157, s)-Nets in Base 2
(157−82, 157, 50)-Net over F2 — Constructive and digital
Digital (75, 157, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(157−82, 157, 160)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 157, 161)-net over F2, because
- 2 times m-reduction [i] would yield digital (75, 155, 161)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2155, 161, F2, 80) (dual of [161, 6, 81]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2156, 162, F2, 80) (dual of [162, 6, 81]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2155, 161, F2, 80) (dual of [161, 6, 81]-code), but
(157−82, 157, 162)-Net in Base 2 — Upper bound on s
There is no (75, 157, 163)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2157, 163, S2, 82), but
- the (dual) Plotkin bound shows that M ≥ 17 538019 647970 835018 444217 992595 396235 871190 515712 / 83 > 2157 [i]