Best Known (212, 212+16, s)-Nets in Base 2
(212, 212+16, 1049087)-Net over F2 — Constructive and digital
Digital (212, 228, 1049087)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (36, 44, 512)-net over F2, using
- net defined by OOA [i] based on linear OOA(244, 512, F2, 8, 8) (dual of [(512, 8), 4052, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, 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(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- net defined by OOA [i] based on linear OOA(244, 512, F2, 8, 8) (dual of [(512, 8), 4052, 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 (36, 44, 512)-net over F2, using
(212, 212+16, 2331689)-Net over F2 — Digital
Digital (212, 228, 2331689)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 2331689, F2, 3, 16) (dual of [(2331689, 3), 6994839, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2228, 2796883, F2, 3, 16) (dual of [(2796883, 3), 8390421, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(244, 682, F2, 3, 8) (dual of [(682, 3), 2002, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(244, 2046, F2, 8) (dual of [2046, 2002, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, 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(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- OOA 3-folding [i] based on linear OA(244, 2046, F2, 8) (dual of [2046, 2002, 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(244, 682, F2, 3, 8) (dual of [(682, 3), 2002, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2228, 2796883, F2, 3, 16) (dual of [(2796883, 3), 8390421, 17]-NRT-code), using
(212, 212+16, large)-Net in Base 2 — Upper bound on s
There is no (212, 228, large)-net in base 2, because
- 14 times m-reduction [i] would yield (212, 214, large)-net in base 2, but