Best Known (252−16, 252, s)-Nets in Base 2
(252−16, 252, 1081343)-Net over F2 — Constructive and digital
Digital (236, 252, 1081343)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (60, 68, 32768)-net over F2, using
- net defined by OOA [i] based on linear OOA(268, 32768, F2, 8, 8) (dual of [(32768, 8), 262076, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- net defined by OOA [i] based on linear OOA(268, 32768, F2, 8, 8) (dual of [(32768, 8), 262076, 9]-NRT-code), 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
- digital (60, 68, 32768)-net over F2, using
(252−16, 252, 3486922)-Net over F2 — Digital
Digital (236, 252, 3486922)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 3486922, F2, 2, 16) (dual of [(3486922, 2), 6973592, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2252, 4259837, F2, 2, 16) (dual of [(4259837, 2), 8519422, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(268, 65536, F2, 2, 8) (dual of [(65536, 2), 131004, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(269, 131073, F2, 9) (dual of [131073, 131004, 10]-code), using
- OOA 2-folding [i] based on linear OA(268, 131072, F2, 8) (dual of [131072, 131004, 9]-code), using
- linear OOA(2184, 4194301, F2, 2, 16) (dual of [(4194301, 2), 8388418, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2184, 8388602, F2, 16) (dual of [8388602, 8388418, 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 2-folding [i] based on linear OA(2184, 8388602, F2, 16) (dual of [8388602, 8388418, 17]-code), using
- linear OOA(268, 65536, F2, 2, 8) (dual of [(65536, 2), 131004, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2252, 4259837, F2, 2, 16) (dual of [(4259837, 2), 8519422, 17]-NRT-code), using
(252−16, 252, large)-Net in Base 2 — Upper bound on s
There is no (236, 252, large)-net in base 2, because
- 14 times m-reduction [i] would yield (236, 238, large)-net in base 2, but