Best Known (211−16, 211, s)-Nets in Base 2
(211−16, 211, 1048625)-Net over F2 — Constructive and digital
Digital (195, 211, 1048625)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 27, 50)-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
(211−16, 211, 2097204)-Net over F2 — Digital
Digital (195, 211, 2097204)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2211, 2097204, F2, 4, 16) (dual of [(2097204, 4), 8388605, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(227, 54, F2, 4, 8) (dual of [(54, 4), 189, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(227, 54, F2, 8) (dual of [54, 27, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(227, 78, F2, 8) (dual of [78, 51, 9]-code), using
- construction XX applied to C1 = C({0,1,3,31}), C2 = C([1,5]), C3 = C1 + C2 = C([1,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,31}) [i] based on
- linear OA(219, 63, F2, 7) (dual of [63, 44, 8]-code), using the primitive cyclic code C(A) with length 63 = 26−1, defining set A = {0,1,3,31}, and minimum distance d ≥ |{−2,−1,…,4}|+1 = 8 (BCH-bound) [i]
- linear OA(218, 63, F2, 6) (dual of [63, 45, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(225, 63, F2, 9) (dual of [63, 38, 10]-code), using the primitive cyclic code C(A) with length 63 = 26−1, defining set A = {0,1,3,5,31}, and minimum distance d ≥ |{−2,−1,…,6}|+1 = 10 (BCH-bound) [i]
- linear OA(212, 63, F2, 4) (dual of [63, 51, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 8, F2, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 7, F2, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C({0,1,3,31}), C2 = C([1,5]), C3 = C1 + C2 = C([1,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,31}) [i] based on
- discarding factors / shortening the dual code based on linear OA(227, 78, F2, 8) (dual of [78, 51, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(227, 54, F2, 8) (dual of [54, 27, 9]-code), using
- linear OOA(2184, 2097150, F2, 4, 16) (dual of [(2097150, 4), 8388416, 17]-NRT-code), using
- OOA 4-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 4-folding [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- linear OOA(227, 54, F2, 4, 8) (dual of [(54, 4), 189, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(211−16, 211, large)-Net in Base 2 — Upper bound on s
There is no (195, 211, large)-net in base 2, because
- 14 times m-reduction [i] would yield (195, 197, large)-net in base 2, but