Best Known (199−95, 199, s)-Nets in Base 2
(199−95, 199, 55)-Net over F2 — Constructive and digital
Digital (104, 199, 55)-net over F2, using
- t-expansion [i] based on digital (100, 199, 55)-net over F2, using
- net from sequence [i] based on digital (100, 54)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 6 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (100, 54)-sequence over F2, using
(199−95, 199, 65)-Net over F2 — Digital
Digital (104, 199, 65)-net over F2, using
- t-expansion [i] based on digital (95, 199, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(199−95, 199, 251)-Net over F2 — Upper bound on s (digital)
There is no digital (104, 199, 252)-net over F2, because
- 3 times m-reduction [i] would yield digital (104, 196, 252)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2196, 252, F2, 92) (dual of [252, 56, 93]-code), but
- construction Y1 [i] would yield
- linear OA(2195, 232, F2, 92) (dual of [232, 37, 93]-code), but
- construction Y1 [i] would yield
- linear OA(2194, 220, F2, 92) (dual of [220, 26, 93]-code), but
- adding a parity check bit [i] would yield linear OA(2195, 221, F2, 93) (dual of [221, 26, 94]-code), but
- OA(237, 232, S2, 12), but
- discarding factors would yield OA(237, 217, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 139173 045698 > 237 [i]
- discarding factors would yield OA(237, 217, S2, 12), but
- linear OA(2194, 220, F2, 92) (dual of [220, 26, 93]-code), but
- construction Y1 [i] would yield
- OA(256, 252, S2, 20), but
- discarding factors would yield OA(256, 224, S2, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 74952 904195 135741 > 256 [i]
- discarding factors would yield OA(256, 224, S2, 20), but
- linear OA(2195, 232, F2, 92) (dual of [232, 37, 93]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2196, 252, F2, 92) (dual of [252, 56, 93]-code), but
(199−95, 199, 276)-Net in Base 2 — Upper bound on s
There is no (104, 199, 277)-net in base 2, because
- 1 times m-reduction [i] would yield (104, 198, 277)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 428508 598675 910414 070491 997122 241031 025726 942110 851224 913024 > 2198 [i]