Best Known (172, 172+17, s)-Nets in Base 2
(172, 172+17, 1048575)-Net over F2 — Constructive and digital
Digital (172, 189, 1048575)-net over F2, using
- 24 times duplication [i] based on 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
(172, 172+17, 1398101)-Net over F2 — Digital
Digital (172, 189, 1398101)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2189, 1398101, F2, 6, 17) (dual of [(1398101, 6), 8388417, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2188, 1398101, F2, 6, 17) (dual of [(1398101, 6), 8388418, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2188, 2796202, F2, 3, 17) (dual of [(2796202, 3), 8388418, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2185, 2796201, F2, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2185, 2796201, F2, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2188, 2796202, F2, 3, 17) (dual of [(2796202, 3), 8388418, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2188, 1398101, F2, 6, 17) (dual of [(1398101, 6), 8388418, 18]-NRT-code), using
(172, 172+17, large)-Net in Base 2 — Upper bound on s
There is no (172, 189, large)-net in base 2, because
- 15 times m-reduction [i] would yield (172, 174, large)-net in base 2, but