Best Known (211, 211+16, s)-Nets in Base 2
(211, 211+16, 1048836)-Net over F2 — Constructive and digital
Digital (211, 227, 1048836)-net over F2, using
- t-expansion [i] based on digital (210, 227, 1048836)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (34, 42, 261)-net over F2, using
- net defined by OOA [i] based on linear OOA(242, 261, F2, 8, 8) (dual of [(261, 8), 2046, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(242, 1044, F2, 8) (dual of [1044, 1002, 9]-code), using
- 1 times truncation [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(221, 1023, F2, 5) (dual of [1023, 1002, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- 1 times truncation [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(242, 1044, F2, 8) (dual of [1044, 1002, 9]-code), using
- net defined by OOA [i] based on linear OOA(242, 261, F2, 8, 8) (dual of [(261, 8), 2046, 9]-NRT-code), using
- digital (168, 185, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (34, 42, 261)-net over F2, using
- (u, u+v)-construction [i] based on
(211, 211+16, 2200821)-Net over F2 — Digital
Digital (211, 227, 2200821)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 2200821, F2, 3, 16) (dual of [(2200821, 3), 6602236, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 2796549, F2, 3, 16) (dual of [(2796549, 3), 8389420, 17]-NRT-code), using
- strength reduction [i] based on linear OOA(2227, 2796549, F2, 3, 17) (dual of [(2796549, 3), 8389420, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(242, 348, F2, 3, 8) (dual of [(348, 3), 1002, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(242, 1044, F2, 8) (dual of [1044, 1002, 9]-code), using
- 1 times truncation [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(221, 1023, F2, 5) (dual of [1023, 1002, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- 1 times truncation [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- OOA 3-folding [i] based on linear OA(242, 1044, F2, 8) (dual of [1044, 1002, 9]-code), using
- linear OOA(2185, 2796201, F2, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- linear OOA(242, 348, F2, 3, 8) (dual of [(348, 3), 1002, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- strength reduction [i] based on linear OOA(2227, 2796549, F2, 3, 17) (dual of [(2796549, 3), 8389420, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 2796549, F2, 3, 16) (dual of [(2796549, 3), 8389420, 17]-NRT-code), using
(211, 211+16, large)-Net in Base 2 — Upper bound on s
There is no (211, 227, large)-net in base 2, because
- 14 times m-reduction [i] would yield (211, 213, large)-net in base 2, but