Best Known (69, 69+66, s)-Nets in Base 2
(69, 69+66, 48)-Net over F2 — Constructive and digital
Digital (69, 135, 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
(69, 69+66, 49)-Net over F2 — Digital
Digital (69, 135, 49)-net over F2, using
- t-expansion [i] based on digital (68, 135, 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
(69, 69+66, 157)-Net over F2 — Upper bound on s (digital)
There is no digital (69, 135, 158)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2135, 158, F2, 66) (dual of [158, 23, 67]-code), but
- construction Y1 [i] would yield
- OA(2134, 150, S2, 66), but
- the linear programming bound shows that M ≥ 235 377396 587616 186439 177776 455843 253082 652672 / 10387 > 2134 [i]
- OA(223, 158, S2, 8), but
- discarding factors would yield OA(223, 120, S2, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 8 502671 > 223 [i]
- discarding factors would yield OA(223, 120, S2, 8), but
- OA(2134, 150, S2, 66), but
- construction Y1 [i] would yield
(69, 69+66, 179)-Net in Base 2 — Upper bound on s
There is no (69, 135, 180)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 45698 069285 363819 257244 564313 769003 352063 > 2135 [i]