Best Known (232, 232+16, s)-Nets in Base 2
(232, 232+16, 1064959)-Net over F2 — Constructive and digital
Digital (232, 248, 1064959)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (56, 64, 16384)-net over F2, using
- net defined by OOA [i] based on linear OOA(264, 16384, F2, 8, 8) (dual of [(16384, 8), 131008, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(264, 65536, F2, 8) (dual of [65536, 65472, 9]-code), using
- 1 times truncation [i] based on linear OA(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(264, 65536, F2, 8) (dual of [65536, 65472, 9]-code), using
- net defined by OOA [i] based on linear OOA(264, 16384, F2, 8, 8) (dual of [(16384, 8), 131008, 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 (56, 64, 16384)-net over F2, using
(232, 232+16, 2818046)-Net over F2 — Digital
Digital (232, 248, 2818046)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 2818046, F2, 3, 16) (dual of [(2818046, 3), 8453890, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(264, 21845, F2, 3, 8) (dual of [(21845, 3), 65471, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(264, 65535, F2, 8) (dual of [65535, 65471, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(264, 65536, F2, 8) (dual of [65536, 65472, 9]-code), using
- 1 times truncation [i] based on linear OA(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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(265, 65537, F2, 9) (dual of [65537, 65472, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(264, 65536, F2, 8) (dual of [65536, 65472, 9]-code), using
- OOA 3-folding [i] based on linear OA(264, 65535, F2, 8) (dual of [65535, 65471, 9]-code), using
- linear OOA(2184, 2796201, F2, 3, 16) (dual of [(2796201, 3), 8388419, 17]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(2184, large, F2, 16) (dual of [large, large−184, 17]-code), using
- linear OOA(264, 21845, F2, 3, 8) (dual of [(21845, 3), 65471, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(232, 232+16, large)-Net in Base 2 — Upper bound on s
There is no (232, 248, large)-net in base 2, because
- 14 times m-reduction [i] would yield (232, 234, large)-net in base 2, but