Best Known (205−16, 205, s)-Nets in Base 2
(205−16, 205, 1048614)-Net over F2 — Constructive and digital
Digital (189, 205, 1048614)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (13, 21, 39)-net over F2, using
- digital (168, 184, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 1048575, F2, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
- OA 8-folding and stacking [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
- OA 8-folding and stacking [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- net defined by OOA [i] based on linear OOA(2184, 1048575, F2, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
(205−16, 205, 1677759)-Net over F2 — Digital
Digital (189, 205, 1677759)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2205, 1677759, F2, 5, 16) (dual of [(1677759, 5), 8388590, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(221, 39, F2, 5, 8) (dual of [(39, 5), 174, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- 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
- linear OOA(221, 39, F2, 5, 8) (dual of [(39, 5), 174, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(205−16, 205, large)-Net in Base 2 — Upper bound on s
There is no (189, 205, large)-net in base 2, because
- 14 times m-reduction [i] would yield (189, 191, large)-net in base 2, but