Best Known (224−17, 224, s)-Nets in Base 2
(224−17, 224, 1048708)-Net over F2 — Constructive and digital
Digital (207, 224, 1048708)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (31, 39, 133)-net over F2, using
- net defined by OOA [i] based on linear OOA(239, 133, F2, 8, 8) (dual of [(133, 8), 1025, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(227, 511, F2, 6) (dual of [511, 484, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(218, 511, F2, 4) (dual of [511, 493, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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, 10, F2, 1) (dual of [10, 9, 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([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- net defined by OOA [i] based on linear OOA(239, 133, F2, 8, 8) (dual of [(133, 8), 1025, 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 (31, 39, 133)-net over F2, using
(224−17, 224, 1746789)-Net over F2 — Digital
Digital (207, 224, 1746789)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2224, 1746789, F2, 4, 17) (dual of [(1746789, 4), 6986932, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2224, 2097283, F2, 4, 17) (dual of [(2097283, 4), 8388908, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2223, 2097283, F2, 4, 17) (dual of [(2097283, 4), 8388909, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2223, 4194566, F2, 2, 17) (dual of [(4194566, 2), 8388909, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(238, 265, F2, 2, 8) (dual of [(265, 2), 492, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(238, 530, F2, 8) (dual of [530, 492, 9]-code), using
- 1 times truncation [i] based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(219, 511, F2, 5) (dual of [511, 492, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- 1 times truncation [i] based on linear OA(239, 531, F2, 9) (dual of [531, 492, 10]-code), using
- OOA 2-folding [i] based on linear OA(238, 530, F2, 8) (dual of [530, 492, 9]-code), using
- linear OOA(2185, 4194301, F2, 2, 17) (dual of [(4194301, 2), 8388417, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2185, 8388602, F2, 17) (dual of [8388602, 8388417, 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 2-folding [i] based on linear OA(2185, 8388602, F2, 17) (dual of [8388602, 8388417, 18]-code), using
- linear OOA(238, 265, F2, 2, 8) (dual of [(265, 2), 492, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA 2-folding [i] based on linear OOA(2223, 4194566, F2, 2, 17) (dual of [(4194566, 2), 8388909, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2223, 2097283, F2, 4, 17) (dual of [(2097283, 4), 8388909, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2224, 2097283, F2, 4, 17) (dual of [(2097283, 4), 8388908, 18]-NRT-code), using
(224−17, 224, large)-Net in Base 2 — Upper bound on s
There is no (207, 224, large)-net in base 2, because
- 15 times m-reduction [i] would yield (207, 209, large)-net in base 2, but