Best Known (219−14, 219, s)-Nets in Base 2
(219−14, 219, 1460513)-Net over F2 — Constructive and digital
Digital (205, 219, 1460513)-net over F2, using
- 21 times duplication [i] based on digital (204, 218, 1460513)-net over F2, using
- t-expansion [i] based on digital (203, 218, 1460513)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (49, 56, 262142)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 262142, F2, 7, 7) (dual of [(262142, 7), 1834938, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(256, 262143, F2, 3, 7) (dual of [(262143, 3), 786373, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(256, 262142, F2, 7, 7) (dual of [(262142, 7), 1834938, 8]-NRT-code), using
- digital (147, 162, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
- digital (49, 56, 262142)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (203, 218, 1460513)-net over F2, using
(219−14, 219, 4261018)-Net over F2 — Digital
Digital (205, 219, 4261018)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 4261018, F2, 2, 14) (dual of [(4261018, 2), 8521817, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 4456445, F2, 2, 14) (dual of [(4456445, 2), 8912671, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2217, 4456444, F2, 2, 14) (dual of [(4456444, 2), 8912671, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(256, 262143, F2, 2, 7) (dual of [(262143, 2), 524230, 8]-NRT-code), using
- linear OOA(2161, 4194301, F2, 2, 14) (dual of [(4194301, 2), 8388441, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OOA 2-folding [i] based on linear OA(2161, 8388602, F2, 14) (dual of [8388602, 8388441, 15]-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2217, 4456444, F2, 2, 14) (dual of [(4456444, 2), 8912671, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 4456445, F2, 2, 14) (dual of [(4456445, 2), 8912671, 15]-NRT-code), using
(219−14, 219, large)-Net in Base 2 — Upper bound on s
There is no (205, 219, large)-net in base 2, because
- 12 times m-reduction [i] would yield (205, 207, large)-net in base 2, but