Best Known (168, 168+13, s)-Nets in Base 2
(168, 168+13, 1406291)-Net over F2 — Constructive and digital
Digital (168, 181, 1406291)-net over F2, using
- 22 times duplication [i] based on digital (166, 179, 1406291)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (34, 40, 8191)-net over F2, using
- 1 times m-reduction [i] based on digital (34, 41, 8191)-net over F2, using
- digital (126, 139, 1398100)-net over F2, using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
- digital (34, 40, 8191)-net over F2, using
- (u, u+v)-construction [i] based on
(168, 168+13, 2804393)-Net over F2 — Digital
Digital (168, 181, 2804393)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2181, 2804393, F2, 3, 13) (dual of [(2804393, 3), 8412998, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(242, 8192, F2, 3, 6) (dual of [(8192, 3), 24534, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 8192, F2, 2, 6) (dual of [(8192, 2), 16342, 7]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(240, 8191, F2, 2, 6) (dual of [(8191, 2), 16342, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (34, 40, 8191)-net over F2, using
- 1 times m-reduction [i] based on digital (34, 41, 8191)-net over F2, using
- extracting embedded OOA [i] based on digital (34, 40, 8191)-net over F2, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(240, 8191, F2, 2, 6) (dual of [(8191, 2), 16342, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 8192, F2, 2, 6) (dual of [(8192, 2), 16342, 7]-NRT-code), using
- linear OOA(2139, 2796201, F2, 3, 13) (dual of [(2796201, 3), 8388464, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- linear OOA(242, 8192, F2, 3, 6) (dual of [(8192, 3), 24534, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(168, 168+13, large)-Net in Base 2 — Upper bound on s
There is no (168, 181, large)-net in base 2, because
- 11 times m-reduction [i] would yield (168, 170, large)-net in base 2, but