Best Known (85, 85+8, s)-Nets in Base 9
(85, 85+8, large)-Net over F9 — Constructive and digital
Digital (85, 93, large)-net over F9, using
- 1 times m-reduction [i] based on digital (85, 94, large)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (21, 25, 4194300)-net over F9, using
- s-reduction based on digital (21, 25, 4194301)-net over F9, using
- net defined by OOA [i] based on linear OOA(925, 4194301, F9, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(925, 4194301, F9, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(925, 8388602, F9, 4) (dual of [8388602, 8388577, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(925, large, F9, 4) (dual of [large, large−25, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(925, large, F9, 4) (dual of [large, large−25, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(925, 8388602, F9, 4) (dual of [8388602, 8388577, 5]-code), using
- appending kth column [i] based on linear OOA(925, 4194301, F9, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(925, 4194301, F9, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- s-reduction based on digital (21, 25, 4194301)-net over F9, using
- digital (57, 66, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- digital (0, 3, 10)-net over F9, using
- generalized (u, u+v)-construction [i] based on
(85, 85+8, large)-Net in Base 9 — Upper bound on s
There is no (85, 93, large)-net in base 9, because
- 6 times m-reduction [i] would yield (85, 87, large)-net in base 9, but