Best Known (64, 64+56, s)-Nets in Base 2
(64, 64+56, 43)-Net over F2 — Constructive and digital
Digital (64, 120, 43)-net over F2, using
- t-expansion [i] based on digital (59, 120, 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
(64, 64+56, 44)-Net over F2 — Digital
Digital (64, 120, 44)-net over F2, using
- t-expansion [i] based on digital (62, 120, 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
(64, 64+56, 161)-Net over F2 — Upper bound on s (digital)
There is no digital (64, 120, 162)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2120, 162, F2, 56) (dual of [162, 42, 57]-code), but
- construction Y1 [i] would yield
- OA(2119, 146, S2, 56), but
- the linear programming bound shows that M ≥ 5165 551323 791927 774620 548045 364260 917630 468096 / 7704 797749 > 2119 [i]
- OA(242, 162, S2, 16), but
- discarding factors would yield OA(242, 146, S2, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 4 467697 956730 > 242 [i]
- discarding factors would yield OA(242, 146, S2, 16), but
- OA(2119, 146, S2, 56), but
- construction Y1 [i] would yield
(64, 64+56, 182)-Net in Base 2 — Upper bound on s
There is no (64, 120, 183)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 492821 419335 931987 267099 221543 545212 > 2120 [i]