Best Known (123, 123+3, s)-Nets in Base 4
(123, 123+3, large)-Net over F4 — Constructive and digital
Digital (123, 126, large)-net over F4, using
- t-expansion [i] based on digital (118, 126, large)-net over F4, using
- trace code for nets [i] based on digital (34, 42, 4194300)-net over F64, using
- net defined by OOA [i] based on linear OOA(6442, 4194300, F64, 10, 8) (dual of [(4194300, 10), 41942958, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6442, 8388601, F64, 2, 8) (dual of [(8388601, 2), 16777160, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6442, 8388602, F64, 2, 8) (dual of [(8388602, 2), 16777162, 9]-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(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-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(6442, 8388602, F64, 2, 8) (dual of [(8388602, 2), 16777162, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6442, 8388601, F64, 2, 8) (dual of [(8388601, 2), 16777160, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6442, 4194300, F64, 10, 8) (dual of [(4194300, 10), 41942958, 9]-NRT-code), using
- trace code for nets [i] based on digital (34, 42, 4194300)-net over F64, using
(123, 123+3, large)-Net in Base 4 — Upper bound on s
There is no (123, 126, large)-net in base 4, because
- 1 times m-reduction [i] would yield (123, 125, large)-net in base 4, but