Best Known (189, 189+12, s)-Nets in Base 2
(189, 189+12, 2446674)-Net over F2 — Constructive and digital
Digital (189, 201, 2446674)-net over F2, using
- 21 times duplication [i] based on digital (188, 200, 2446674)-net over F2, using
- t-expansion [i] based on digital (187, 200, 2446674)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (55, 61, 1048574)-net over F2, using
- 1 times m-reduction [i] based on digital (55, 62, 1048574)-net over F2, using
- net defined by OOA [i] based on linear OOA(262, 1048574, F2, 7, 7) (dual of [(1048574, 7), 7339956, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(262, 1048575, F2, 3, 7) (dual of [(1048575, 3), 3145663, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(262, 1048574, F2, 7, 7) (dual of [(1048574, 7), 7339956, 8]-NRT-code), using
- 1 times m-reduction [i] based on digital (55, 62, 1048574)-net over F2, using
- digital (126, 139, 1398100)-net over F2, using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
- digital (55, 61, 1048574)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (187, 200, 2446674)-net over F2, using
(189, 189+12, 5242877)-Net over F2 — Digital
Digital (189, 201, 5242877)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 5242877, F2, 2, 12) (dual of [(5242877, 2), 10485553, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2199, 5242876, F2, 2, 12) (dual of [(5242876, 2), 10485553, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(261, 1048575, F2, 2, 6) (dual of [(1048575, 2), 2097089, 7]-NRT-code), using
- linear OOA(2138, 4194301, F2, 2, 12) (dual of [(4194301, 2), 8388464, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2138, 8388602, F2, 12) (dual of [8388602, 8388464, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- OOA 2-folding [i] based on linear OA(2138, 8388602, F2, 12) (dual of [8388602, 8388464, 13]-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2199, 5242876, F2, 2, 12) (dual of [(5242876, 2), 10485553, 13]-NRT-code), using
(189, 189+12, large)-Net in Base 2 — Upper bound on s
There is no (189, 201, large)-net in base 2, because
- 10 times m-reduction [i] would yield (189, 191, large)-net in base 2, but