Best Known (186−16, 186, s)-Nets in Base 2
(186−16, 186, 1048575)-Net over F2 — Constructive and digital
Digital (170, 186, 1048575)-net over F2, using
- 21 times duplication [i] based on digital (169, 185, 1048575)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (168, 185, 1048575)-net over F2, using
(186−16, 186, 1398100)-Net over F2 — Digital
Digital (170, 186, 1398100)-net over F2, using
- 22 times duplication [i] based on digital (168, 184, 1398100)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2184, 1398100, F2, 6, 16) (dual of [(1398100, 6), 8388416, 17]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, large, F2, 16) (dual of [large, large−184, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2184, large, F2, 16) (dual of [large, large−184, 17]-code), using
- OOA 6-folding [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2184, 1398100, F2, 6, 16) (dual of [(1398100, 6), 8388416, 17]-NRT-code), using
(186−16, 186, large)-Net in Base 2 — Upper bound on s
There is no (170, 186, large)-net in base 2, because
- 14 times m-reduction [i] would yield (170, 172, large)-net in base 2, but