Best Known (117−54, 117, s)-Nets in Base 2
(117−54, 117, 43)-Net over F2 — Constructive and digital
Digital (63, 117, 43)-net over F2, using
- t-expansion [i] based on digital (59, 117, 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
(117−54, 117, 44)-Net over F2 — Digital
Digital (63, 117, 44)-net over F2, using
- t-expansion [i] based on digital (62, 117, 44)-net over F2, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
(117−54, 117, 167)-Net over F2 — Upper bound on s (digital)
There is no digital (63, 117, 168)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2117, 168, F2, 54) (dual of [168, 51, 55]-code), but
- construction Y1 [i] would yield
- OA(2116, 148, S2, 54), but
- the linear programming bound shows that M ≥ 67 089316 460630 621624 223128 238837 181334 945792 / 674 571975 > 2116 [i]
- OA(251, 168, S2, 20), but
- discarding factors would yield OA(251, 159, S2, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 2282 987168 569557 > 251 [i]
- discarding factors would yield OA(251, 159, S2, 20), but
- OA(2116, 148, S2, 54), but
- construction Y1 [i] would yield
(117−54, 117, 183)-Net in Base 2 — Upper bound on s
There is no (63, 117, 184)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182467 222295 912684 213142 696981 078356 > 2117 [i]