Best Known (229, 229+12, s)-Nets in Base 2
(229, 229+12, 2804388)-Net over F2 — Constructive and digital
Digital (229, 241, 2804388)-net over F2, using
- 21 times duplication [i] based on digital (228, 240, 2804388)-net over F2, using
- t-expansion [i] based on digital (227, 240, 2804388)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 64, 2097150)-net over F2, using
- 1 times m-reduction [i] based on digital (58, 65, 2097150)-net over F2, using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(265, 2097151, F2, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(265, 2097150, F2, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
- 1 times m-reduction [i] based on digital (58, 65, 2097150)-net over F2, using
- digital (163, 176, 1402194)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (31, 37, 4094)-net over F2, using
- 1 times m-reduction [i] based on digital (31, 38, 4094)-net over F2, using
- net defined by OOA [i] based on linear OOA(238, 4094, F2, 7, 7) (dual of [(4094, 7), 28620, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(238, 4095, F2, 3, 7) (dual of [(4095, 3), 12247, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(238, 4094, F2, 7, 7) (dual of [(4094, 7), 28620, 8]-NRT-code), using
- 1 times m-reduction [i] based on digital (31, 38, 4094)-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 (31, 37, 4094)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (58, 64, 2097150)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (227, 240, 2804388)-net over F2, using
(229, 229+12, large)-Net over F2 — Digital
Digital (229, 241, large)-net over F2, using
- 210 times duplication [i] based on digital (219, 231, large)-net over F2, using
- t-expansion [i] based on digital (217, 231, large)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, large, F2, 2, 14), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2229, 8388602, F2, 2, 14) (dual of [(8388602, 2), 16776975, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(268, 4194303, F2, 2, 7) (dual of [(4194303, 2), 8388538, 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(2229, 8388602, F2, 2, 14) (dual of [(8388602, 2), 16776975, 15]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, large, F2, 2, 14), using
- t-expansion [i] based on digital (217, 231, large)-net over F2, using
(229, 229+12, large)-Net in Base 2 — Upper bound on s
There is no (229, 241, large)-net in base 2, because
- 10 times m-reduction [i] would yield (229, 231, large)-net in base 2, but