Best Known (34, 34+32, s)-Nets in Base 2
(34, 34+32, 24)-Net over F2 — Constructive and digital
Digital (34, 66, 24)-net over F2, using
- t-expansion [i] based on digital (33, 66, 24)-net over F2, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 24, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
(34, 34+32, 28)-Net over F2 — Digital
Digital (34, 66, 28)-net over F2, using
- t-expansion [i] based on digital (33, 66, 28)-net over F2, using
- net from sequence [i] based on digital (33, 27)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 28, using
- net from sequence [i] based on digital (33, 27)-sequence over F2, using
(34, 34+32, 81)-Net over F2 — Upper bound on s (digital)
There is no digital (34, 66, 82)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(266, 82, F2, 32) (dual of [82, 16, 33]-code), but
- construction Y1 [i] would yield
- linear OA(265, 76, F2, 32) (dual of [76, 11, 33]-code), but
- construction Y1 [i] would yield
- linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- OA(211, 76, S2, 4), but
- discarding factors would yield OA(211, 64, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 2081 > 211 [i]
- discarding factors would yield OA(211, 64, S2, 4), but
- linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- construction Y1 [i] would yield
- OA(216, 82, S2, 6), but
- discarding factors would yield OA(216, 74, S2, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 67600 > 216 [i]
- discarding factors would yield OA(216, 74, S2, 6), but
- linear OA(265, 76, F2, 32) (dual of [76, 11, 33]-code), but
- construction Y1 [i] would yield
(34, 34+32, 84)-Net in Base 2 — Upper bound on s
There is no (34, 66, 85)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(266, 85, S2, 32), but
- the linear programming bound shows that M ≥ 205 069945 110235 060816 052224 / 2 773125 > 266 [i]