Best Known (89−8, 89, s)-Nets in Base 4
(89−8, 89, 8388600)-Net over F4 — Constructive and digital
Digital (81, 89, 8388600)-net over F4, using
- 41 times duplication [i] based on digital (80, 88, 8388600)-net over F4, using
- trace code for nets [i] based on digital (36, 44, 4194300)-net over F16, using
- net defined by OOA [i] based on linear OOA(1644, 4194300, F16, 10, 8) (dual of [(4194300, 10), 41942956, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1644, 8388601, F16, 2, 8) (dual of [(8388601, 2), 16777158, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1644, 8388602, F16, 2, 8) (dual of [(8388602, 2), 16777160, 9]-NRT-code), using
- trace code [i] based on linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- trace code [i] based on linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1644, 8388602, F16, 2, 8) (dual of [(8388602, 2), 16777160, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1644, 8388601, F16, 2, 8) (dual of [(8388601, 2), 16777158, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1644, 4194300, F16, 10, 8) (dual of [(4194300, 10), 41942956, 9]-NRT-code), using
- trace code for nets [i] based on digital (36, 44, 4194300)-net over F16, using
(89−8, 89, large)-Net over F4 — Digital
Digital (81, 89, large)-net over F4, using
- 48 times duplication [i] based on digital (73, 81, large)-net over F4, using
- t-expansion [i] based on digital (72, 81, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(481, large, F4, 9) (dual of [large, large−81, 10]-code), using
- 8 times code embedding in larger space [i] based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 8 times code embedding in larger space [i] based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(481, large, F4, 9) (dual of [large, large−81, 10]-code), using
- t-expansion [i] based on digital (72, 81, large)-net over F4, using
(89−8, 89, large)-Net in Base 4 — Upper bound on s
There is no (81, 89, large)-net in base 4, because
- 6 times m-reduction [i] would yield (81, 83, large)-net in base 4, but