Best Known (224−27, 224, s)-Nets in Base 2
(224−27, 224, 10083)-Net over F2 — Constructive and digital
Digital (197, 224, 10083)-net over F2, using
- 21 times duplication [i] based on digital (196, 223, 10083)-net over F2, using
- net defined by OOA [i] based on linear OOA(2223, 10083, F2, 27, 27) (dual of [(10083, 27), 272018, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2223, 131080, F2, 27) (dual of [131080, 130857, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2223, 131090, F2, 27) (dual of [131090, 130867, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2222, 131072, F2, 27) (dual of [131072, 130850, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2205, 131072, F2, 25) (dual of [131072, 130867, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2223, 131090, F2, 27) (dual of [131090, 130867, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2223, 131080, F2, 27) (dual of [131080, 130857, 28]-code), using
- net defined by OOA [i] based on linear OOA(2223, 10083, F2, 27, 27) (dual of [(10083, 27), 272018, 28]-NRT-code), using
(224−27, 224, 18727)-Net over F2 — Digital
Digital (197, 224, 18727)-net over F2, using
- 21 times duplication [i] based on digital (196, 223, 18727)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 18727, F2, 7, 27) (dual of [(18727, 7), 130866, 28]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2223, 131089, F2, 27) (dual of [131089, 130866, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2223, 131090, F2, 27) (dual of [131090, 130867, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2222, 131072, F2, 27) (dual of [131072, 130850, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2205, 131072, F2, 25) (dual of [131072, 130867, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2223, 131090, F2, 27) (dual of [131090, 130867, 28]-code), using
- OOA 7-folding [i] based on linear OA(2223, 131089, F2, 27) (dual of [131089, 130866, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 18727, F2, 7, 27) (dual of [(18727, 7), 130866, 28]-NRT-code), using
(224−27, 224, 826453)-Net in Base 2 — Upper bound on s
There is no (197, 224, 826454)-net in base 2, because
- 1 times m-reduction [i] would yield (197, 223, 826454)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13 480077 678892 161240 969533 011300 867225 327980 273469 096293 977958 677502 > 2223 [i]