Best Known (216, 216+17, s)-Nets in Base 2
(216, 216+17, 1049599)-Net over F2 — Constructive and digital
Digital (216, 233, 1049599)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (40, 48, 1024)-net over F2, using
- net defined by OOA [i] based on linear OOA(248, 1024, F2, 8, 8) (dual of [(1024, 8), 8144, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(248, 4096, F2, 8) (dual of [4096, 4048, 9]-code), using
- 1 times truncation [i] based on linear OA(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, 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(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(248, 4096, F2, 8) (dual of [4096, 4048, 9]-code), using
- net defined by OOA [i] based on linear OOA(248, 1024, F2, 8, 8) (dual of [(1024, 8), 8144, 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 (40, 48, 1024)-net over F2, using
(216, 216+17, 2098676)-Net over F2 — Digital
Digital (216, 233, 2098676)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 2098676, F2, 4, 17) (dual of [(2098676, 4), 8394471, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(248, 1526, F2, 4, 8) (dual of [(1526, 4), 6056, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(248, 1526, F2, 2, 8) (dual of [(1526, 2), 3004, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(248, 2048, F2, 2, 8) (dual of [(2048, 2), 4048, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(248, 4096, F2, 8) (dual of [4096, 4048, 9]-code), using
- 1 times truncation [i] based on linear OA(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, 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(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- OOA 2-folding [i] based on linear OA(248, 4096, F2, 8) (dual of [4096, 4048, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(248, 2048, F2, 2, 8) (dual of [(2048, 2), 4048, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(248, 1526, F2, 2, 8) (dual of [(1526, 2), 3004, 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(248, 1526, F2, 4, 8) (dual of [(1526, 4), 6056, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(216, 216+17, large)-Net in Base 2 — Upper bound on s
There is no (216, 233, large)-net in base 2, because
- 15 times m-reduction [i] would yield (216, 218, large)-net in base 2, but