Best Known (157−78, 157, s)-Nets in Base 2
(157−78, 157, 50)-Net over F2 — Constructive and digital
Digital (79, 157, 50)-net over F2, using
- t-expansion [i] based on 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]
- net from sequence [i] based on digital (75, 49)-sequence over F2, using
(157−78, 157, 52)-Net over F2 — Digital
Digital (79, 157, 52)-net over F2, using
- t-expansion [i] based on digital (77, 157, 52)-net over F2, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 77 and N(F) ≥ 52, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
(157−78, 157, 175)-Net over F2 — Upper bound on s (digital)
There is no digital (79, 157, 176)-net over F2, because
- 4 times m-reduction [i] would yield digital (79, 153, 176)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2153, 176, F2, 74) (dual of [176, 23, 75]-code), but
- 2 times code embedding in larger space [i] would yield linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- adding a parity check bit [i] would yield linear OA(2156, 179, F2, 75) (dual of [179, 23, 76]-code), but
- 2 times code embedding in larger space [i] would yield linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2153, 176, F2, 74) (dual of [176, 23, 75]-code), but
(157−78, 157, 198)-Net in Base 2 — Upper bound on s
There is no (79, 157, 199)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 206869 139282 005388 028388 803748 319138 912587 796028 > 2157 [i]