Best Known (172−4, 172, s)-Nets in Base 3
(172−4, 172, large)-Net over F3 — Constructive and digital
Digital (168, 172, large)-net over F3, using
- 31 times duplication [i] based on digital (167, 171, large)-net over F3, using
- t-expansion [i] based on digital (162, 171, large)-net over F3, using
- trace code for nets [i] based on digital (48, 57, 4194300)-net over F27, using
- net defined by OOA [i] based on linear OOA(2757, 4194300, F27, 10, 9) (dual of [(4194300, 10), 41942943, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2757, 8388601, F27, 2, 9) (dual of [(8388601, 2), 16777145, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2757, 8388602, F27, 2, 9) (dual of [(8388602, 2), 16777147, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2716, 4194301, F27, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- OOA 2-folding [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- linear OOA(2741, 4194301, F27, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2741, 8388602, F27, 9) (dual of [8388602, 8388561, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- OOA 2-folding [i] based on linear OA(2741, 8388602, F27, 9) (dual of [8388602, 8388561, 10]-code), using
- linear OOA(2716, 4194301, F27, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2757, 8388602, F27, 2, 9) (dual of [(8388602, 2), 16777147, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2757, 8388601, F27, 2, 9) (dual of [(8388601, 2), 16777145, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2757, 4194300, F27, 10, 9) (dual of [(4194300, 10), 41942943, 10]-NRT-code), using
- trace code for nets [i] based on digital (48, 57, 4194300)-net over F27, using
- t-expansion [i] based on digital (162, 171, large)-net over F3, using
(172−4, 172, large)-Net in Base 3 — Upper bound on s
There is no (168, 172, large)-net in base 3, because
- 2 times m-reduction [i] would yield (168, 170, large)-net in base 3, but