Best Known (191−16, 191, s)-Nets in Base 2
(191−16, 191, 1048575)-Net over F2 — Constructive and digital
Digital (175, 191, 1048575)-net over F2, using
- 26 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
(191−16, 191, 1677721)-Net over F2 — Digital
Digital (175, 191, 1677721)-net over F2, using
- 21 times duplication [i] based on digital (174, 190, 1677721)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2190, 1677721, F2, 5, 16) (dual of [(1677721, 5), 8388415, 17]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2189, 1677721, F2, 5, 16) (dual of [(1677721, 5), 8388416, 17]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2184, 1677720, F2, 5, 16) (dual of [(1677720, 5), 8388416, 17]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2184, 1677720, F2, 5, 16) (dual of [(1677720, 5), 8388416, 17]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2189, 1677721, F2, 5, 16) (dual of [(1677721, 5), 8388416, 17]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2190, 1677721, F2, 5, 16) (dual of [(1677721, 5), 8388415, 17]-NRT-code), using
(191−16, 191, large)-Net in Base 2 — Upper bound on s
There is no (175, 191, large)-net in base 2, because
- 14 times m-reduction [i] would yield (175, 177, large)-net in base 2, but