Best Known (214, 214+13, s)-Nets in Base 2
(214, 214+13, 2796460)-Net over F2 — Constructive and digital
Digital (214, 227, 2796460)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 64, 2097150)-net over F2, using
- 1 times m-reduction [i] based on digital (58, 65, 2097150)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(265, 2097151, F2, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- 1 times m-reduction [i] based on digital (58, 65, 2097150)-net over F2, using
- digital (150, 163, 1398230)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (18, 24, 130)-net over F2, using
- net defined by OOA [i] based on linear OOA(224, 130, F2, 6, 6) (dual of [(130, 6), 756, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(224, 130, F2, 5, 6) (dual of [(130, 5), 626, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(23, 3, F2, 5, 3) (dual of [(3, 5), 12, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(5;12,2) [i]
- linear OOA(221, 127, F2, 5, 6) (dual of [(127, 5), 614, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (15, 21, 127)-net over F2, using
- linear OOA(23, 3, F2, 5, 3) (dual of [(3, 5), 12, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(224, 130, F2, 5, 6) (dual of [(130, 5), 626, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(224, 130, F2, 6, 6) (dual of [(130, 6), 756, 7]-NRT-code), 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 (18, 24, 130)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 64, 2097150)-net over F2, using
(214, 214+13, large)-Net over F2 — Digital
Digital (214, 227, large)-net over F2, using
- 216 times duplication [i] based on digital (198, 211, large)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2211, large, F2, 2, 13), using
- 21 times duplication [i] based on linear OOA(2210, large, F2, 2, 13), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2206, 8388602, F2, 2, 13) (dual of [(8388602, 2), 16776998, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(267, 4194303, F2, 2, 6) (dual of [(4194303, 2), 8388539, 7]-NRT-code), using
- linear OOA(2139, 4194301, F2, 2, 13) (dual of [(4194301, 2), 8388463, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2139, 8388602, F2, 13) (dual of [8388602, 8388463, 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 2-folding [i] based on linear OA(2139, 8388602, F2, 13) (dual of [8388602, 8388463, 14]-code), using
- (u, u+v)-construction [i] based on
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2206, 8388602, F2, 2, 13) (dual of [(8388602, 2), 16776998, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2210, large, F2, 2, 13), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2211, large, F2, 2, 13), using
(214, 214+13, large)-Net in Base 2 — Upper bound on s
There is no (214, 227, large)-net in base 2, because
- 11 times m-reduction [i] would yield (214, 216, large)-net in base 2, but