Best Known (105, 105+92, s)-Nets in Base 2
(105, 105+92, 56)-Net over F2 — Constructive and digital
Digital (105, 197, 56)-net over F2, using
- net from sequence [i] based on digital (105, 55)-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 7 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]
(105, 105+92, 65)-Net over F2 — Digital
Digital (105, 197, 65)-net over F2, using
- t-expansion [i] based on digital (95, 197, 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
(105, 105+92, 283)-Net over F2 — Upper bound on s (digital)
There is no digital (105, 197, 284)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2197, 284, F2, 92) (dual of [284, 87, 93]-code), but
- adding a parity check bit [i] would yield linear OA(2198, 285, F2, 93) (dual of [285, 87, 94]-code), but
- construction Y1 [i] would yield
- linear OA(2197, 253, F2, 93) (dual of [253, 56, 94]-code), but
- 1 times truncation [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
- 1 times truncation [i] would yield linear OA(2196, 252, F2, 92) (dual of [252, 56, 93]-code), but
- linear OA(287, 285, F2, 32) (dual of [285, 198, 33]-code), but
- the improved Johnson bound shows that N ≤ 614947 375600 444218 516740 363509 225653 692016 328721 907942 627066 < 2198 [i]
- linear OA(2197, 253, F2, 93) (dual of [253, 56, 94]-code), but
- construction Y1 [i] would yield
- adding a parity check bit [i] would yield linear OA(2198, 285, F2, 93) (dual of [285, 87, 94]-code), but
(105, 105+92, 284)-Net in Base 2 — Upper bound on s
There is no (105, 197, 285)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2197, 285, S2, 92), but
- 15 times code embedding in larger space [i] would yield OA(2212, 300, S2, 92), but
- the linear programming bound shows that M ≥ 2484 132694 351399 471861 447697 940992 402712 231248 004226 234322 576635 442083 069406 167011 327832 607273 517056 / 377277 818302 855799 777736 052112 623275 > 2212 [i]
- 15 times code embedding in larger space [i] would yield OA(2212, 300, S2, 92), but