Best Known (224, 224+17, s)-Nets in Base 2
(224, 224+17, 1052671)-Net over F2 — Constructive and digital
Digital (224, 241, 1052671)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (48, 56, 4096)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 4096, F2, 8, 8) (dual of [(4096, 8), 32712, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- net defined by OOA [i] based on linear OOA(256, 4096, F2, 8, 8) (dual of [(4096, 8), 32712, 9]-NRT-code), using
- digital (168, 185, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (48, 56, 4096)-net over F2, using
(224, 224+17, 2102611)-Net over F2 — Digital
Digital (224, 241, 2102611)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 2102611, F2, 4, 17) (dual of [(2102611, 4), 8410203, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(256, 5461, F2, 4, 8) (dual of [(5461, 4), 21788, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 5461, F2, 3, 8) (dual of [(5461, 3), 16327, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(256, 16383, F2, 8) (dual of [16383, 16327, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(257, 16385, F2, 9) (dual of [16385, 16328, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(256, 16384, F2, 8) (dual of [16384, 16328, 9]-code), using
- OOA 3-folding [i] based on linear OA(256, 16383, F2, 8) (dual of [16383, 16327, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 5461, F2, 3, 8) (dual of [(5461, 3), 16327, 9]-NRT-code), using
- linear OOA(2185, 2097150, F2, 4, 17) (dual of [(2097150, 4), 8388415, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 4-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- linear OOA(256, 5461, F2, 4, 8) (dual of [(5461, 4), 21788, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(224, 224+17, large)-Net in Base 2 — Upper bound on s
There is no (224, 241, large)-net in base 2, because
- 15 times m-reduction [i] would yield (224, 226, large)-net in base 2, but