Best Known (174, 186, s)-Nets in Base 2
(174, 186, 1430866)-Net over F2 — Constructive and digital
Digital (174, 186, 1430866)-net over F2, using
- 21 times duplication [i] based on digital (173, 185, 1430866)-net over F2, using
- t-expansion [i] based on digital (172, 185, 1430866)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (40, 46, 32766)-net over F2, using
- 1 times m-reduction [i] based on digital (40, 47, 32766)-net over F2, using
- net defined by OOA [i] based on linear OOA(247, 32766, F2, 7, 7) (dual of [(32766, 7), 229315, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(247, 32767, F2, 3, 7) (dual of [(32767, 3), 98254, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(247, 32766, F2, 7, 7) (dual of [(32766, 7), 229315, 8]-NRT-code), using
- 1 times m-reduction [i] based on digital (40, 47, 32766)-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 (40, 46, 32766)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (172, 185, 1430866)-net over F2, using
(174, 186, 4227069)-Net over F2 — Digital
Digital (174, 186, 4227069)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2186, 4227069, F2, 2, 12) (dual of [(4227069, 2), 8453952, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2184, 4227068, F2, 2, 12) (dual of [(4227068, 2), 8453952, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(246, 32767, F2, 2, 6) (dual of [(32767, 2), 65488, 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(2184, 4227068, F2, 2, 12) (dual of [(4227068, 2), 8453952, 13]-NRT-code), using
(174, 186, large)-Net in Base 2 — Upper bound on s
There is no (174, 186, large)-net in base 2, because
- 10 times m-reduction [i] would yield (174, 176, large)-net in base 2, but