Best Known (137−8, 137, s)-Nets in Base 4
(137−8, 137, large)-Net over F4 — Constructive and digital
Digital (129, 137, large)-net over F4, using
- 1 times m-reduction [i] based on digital (129, 138, large)-net over F4, using
- trace code for nets [i] based on digital (37, 46, 4194300)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 4194300, F64, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6446, 8388601, F64, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6446, 8388602, F64, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(6413, 4194301, F64, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- linear OOA(6433, 4194301, F64, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- linear OOA(6413, 4194301, F64, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(6446, 8388602, F64, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6446, 8388601, F64, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6446, 4194300, F64, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- trace code for nets [i] based on digital (37, 46, 4194300)-net over F64, using
(137−8, 137, large)-Net in Base 4 — Upper bound on s
There is no (129, 137, large)-net in base 4, because
- 6 times m-reduction [i] would yield (129, 131, large)-net in base 4, but