Best Known (84−9, 84, s)-Nets in Base 9
(84−9, 84, 4223837)-Net over F9 — Constructive and digital
Digital (75, 84, 4223837)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (14, 18, 29537)-net over F9, using
- net defined by OOA [i] based on linear OOA(918, 29537, F9, 4, 4) (dual of [(29537, 4), 118130, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(918, 29537, F9, 3, 4) (dual of [(29537, 3), 88593, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(92, 10, F9, 3, 2) (dual of [(10, 3), 28, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;28,9) [i]
- linear OOA(916, 29527, F9, 3, 4) (dual of [(29527, 3), 88565, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 59049, F9, 3) (dual of [59049, 59038, 4]-code or 59049-cap in PG(10,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- linear OOA(92, 10, F9, 3, 2) (dual of [(10, 3), 28, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(918, 29537, F9, 3, 4) (dual of [(29537, 3), 88593, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(918, 29537, F9, 4, 4) (dual of [(29537, 4), 118130, 5]-NRT-code), 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 (14, 18, 29537)-net over F9, using
(84−9, 84, large)-Net over F9 — Digital
Digital (75, 84, large)-net over F9, using
- 93 times duplication [i] based on digital (72, 81, large)-net over F9, using
- t-expansion [i] based on digital (69, 81, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- t-expansion [i] based on digital (69, 81, large)-net over F9, using
(84−9, 84, large)-Net in Base 9 — Upper bound on s
There is no (75, 84, large)-net in base 9, because
- 7 times m-reduction [i] would yield (75, 77, large)-net in base 9, but