Best Known (133−64, 133, s)-Nets in Base 2
(133−64, 133, 48)-Net over F2 — Constructive and digital
Digital (69, 133, 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
(133−64, 133, 49)-Net over F2 — Digital
Digital (69, 133, 49)-net over F2, using
- t-expansion [i] based on digital (68, 133, 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
(133−64, 133, 159)-Net over F2 — Upper bound on s (digital)
There is no digital (69, 133, 160)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2133, 160, F2, 64) (dual of [160, 27, 65]-code), but
- construction Y1 [i] would yield
- OA(2132, 150, S2, 64), but
- the linear programming bound shows that M ≥ 7654 730789 395636 393332 135297 086550 086927 777792 / 1 285141 > 2132 [i]
- OA(227, 160, S2, 10), but
- discarding factors would yield OA(227, 111, S2, 10), but
- the Rao or (dual) Hamming bound shows that M ≥ 134 381744 > 227 [i]
- discarding factors would yield OA(227, 111, S2, 10), but
- OA(2132, 150, S2, 64), but
- construction Y1 [i] would yield
(133−64, 133, 184)-Net in Base 2 — Upper bound on s
There is no (69, 133, 185)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11504 281992 353378 611645 981790 814358 209807 > 2133 [i]