Best Known (68, 128, s)-Nets in Base 2
(68, 128, 43)-Net over F2 — Constructive and digital
Digital (68, 128, 43)-net over F2, using
- t-expansion [i] based on digital (59, 128, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(68, 128, 49)-Net over F2 — Digital
Digital (68, 128, 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
(68, 128, 180)-Net over F2 — Upper bound on s (digital)
There is no digital (68, 128, 181)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2128, 181, F2, 60) (dual of [181, 53, 61]-code), but
- construction Y1 [i] would yield
- linear OA(2127, 161, F2, 60) (dual of [161, 34, 61]-code), but
- construction Y1 [i] would yield
- OA(2126, 149, S2, 60), but
- the linear programming bound shows that M ≥ 40 354766 457887 934259 048521 444548 255732 989952 / 438495 > 2126 [i]
- OA(234, 161, S2, 12), but
- discarding factors would yield OA(234, 154, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 17486 314616 > 234 [i]
- discarding factors would yield OA(234, 154, S2, 12), but
- OA(2126, 149, S2, 60), but
- construction Y1 [i] would yield
- OA(253, 181, S2, 20), but
- discarding factors would yield OA(253, 180, S2, 20), but
- the linear programming bound shows that M ≥ 2 488754 712874 464491 680229 064205 986311 441557 225472 / 271 456013 974306 311726 277255 439899 > 253 [i]
- discarding factors would yield OA(253, 180, S2, 20), but
- linear OA(2127, 161, F2, 60) (dual of [161, 34, 61]-code), but
- construction Y1 [i] would yield
(68, 128, 190)-Net in Base 2 — Upper bound on s
There is no (68, 128, 191)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 346 156150 426799 456837 861778 160199 449695 > 2128 [i]