Best Known (209−17, 209, s)-Nets in Base 2
(209−17, 209, 1048614)-Net over F2 — Constructive and digital
Digital (192, 209, 1048614)-net over F2, using
- 23 times duplication [i] based on digital (189, 206, 1048614)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (13, 21, 39)-net over F2, 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
- (u, u+v)-construction [i] based on
(209−17, 209, 1677760)-Net over F2 — Digital
Digital (192, 209, 1677760)-net over F2, using
- 21 times duplication [i] based on digital (191, 208, 1677760)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 1677760, F2, 5, 17) (dual of [(1677760, 5), 8388592, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(223, 40, F2, 5, 8) (dual of [(40, 5), 177, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(223, 40, F2, 2, 8) (dual of [(40, 2), 57, 9]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(221, 39, F2, 2, 8) (dual of [(39, 2), 57, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(221, 39, F2, 2, 8) (dual of [(39, 2), 57, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(223, 40, F2, 2, 8) (dual of [(40, 2), 57, 9]-NRT-code), using
- linear OOA(2185, 1677720, F2, 5, 17) (dual of [(1677720, 5), 8388415, 18]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- linear OOA(223, 40, F2, 5, 8) (dual of [(40, 5), 177, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 1677760, F2, 5, 17) (dual of [(1677760, 5), 8388592, 18]-NRT-code), using
(209−17, 209, large)-Net in Base 2 — Upper bound on s
There is no (192, 209, large)-net in base 2, because
- 15 times m-reduction [i] would yield (192, 194, large)-net in base 2, but