Best Known (69−9, 69, s)-Nets in Base 9
(69−9, 69, 4194301)-Net over F9 — Constructive and digital
Digital (60, 69, 4194301)-net over F9, using
- 91 times duplication [i] based on digital (59, 68, 4194301)-net over F9, using
- net defined by OOA [i] based on linear OOA(968, 4194301, F9, 10, 9) (dual of [(4194301, 10), 41942942, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(968, large, F9, 2, 9), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, 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(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(968, large, F9, 2, 9), using
- net defined by OOA [i] based on linear OOA(968, 4194301, F9, 10, 9) (dual of [(4194301, 10), 41942942, 10]-NRT-code), using
(69−9, 69, large)-Net over F9 — Digital
Digital (60, 69, large)-net over F9, using
- 94 times duplication [i] based on digital (56, 65, large)-net over F9, using
- t-expansion [i] based on digital (55, 65, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- t-expansion [i] based on digital (55, 65, large)-net over F9, using
(69−9, 69, large)-Net in Base 9 — Upper bound on s
There is no (60, 69, large)-net in base 9, because
- 7 times m-reduction [i] would yield (60, 62, large)-net in base 9, but