Best Known (75−9, 75, s)-Nets in Base 9
(75−9, 75, 4194464)-Net over F9 — Constructive and digital
Digital (66, 75, 4194464)-net over F9, using
- 91 times duplication [i] based on digital (65, 74, 4194464)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, 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 (4, 8, 164)-net over F9, using
- (u, u+v)-construction [i] based on
(75−9, 75, 4194651)-Net in Base 9 — Constructive
(66, 75, 4194651)-net in base 9, using
- (u, u+v)-construction [i] based on
- (5, 9, 351)-net in base 9, using
- base change [i] based on digital (2, 6, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(276, 351, F27, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(276, 702, F27, 4) (dual of [702, 696, 5]-code), using
- 1 times truncation [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(276, 702, F27, 4) (dual of [702, 696, 5]-code), using
- net defined by OOA [i] based on linear OOA(276, 351, F27, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 351)-net over F27, 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
- (5, 9, 351)-net in base 9, using
(75−9, 75, large)-Net over F9 — Digital
Digital (66, 75, large)-net over F9, using
- 92 times duplication [i] based on digital (64, 73, large)-net over F9, using
- t-expansion [i] based on digital (62, 73, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(973, large, F9, 11) (dual of [large, large−73, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(973, large, F9, 11) (dual of [large, large−73, 12]-code), using
- t-expansion [i] based on digital (62, 73, large)-net over F9, using
(75−9, 75, large)-Net in Base 9 — Upper bound on s
There is no (66, 75, large)-net in base 9, because
- 7 times m-reduction [i] would yield (66, 68, large)-net in base 9, but