Best Known (77, 77+9, s)-Nets in Base 9
(77, 77+9, 4725744)-Net over F9 — Constructive and digital
Digital (77, 86, 4725744)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (16, 20, 531444)-net over F9, using
- net defined by OOA [i] based on linear OOA(920, 531444, F9, 4, 4) (dual of [(531444, 4), 2125756, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(920, 531444, F9, 3, 4) (dual of [(531444, 3), 1594312, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(920, 1062888, F9, 4) (dual of [1062888, 1062868, 5]-code), using
- trace code [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(817, 531441, F81, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(920, 1062888, F9, 4) (dual of [1062888, 1062868, 5]-code), using
- appending kth column [i] based on linear OOA(920, 531444, F9, 3, 4) (dual of [(531444, 3), 1594312, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(920, 531444, F9, 4, 4) (dual of [(531444, 4), 2125756, 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 (16, 20, 531444)-net over F9, using
(77, 77+9, large)-Net over F9 — Digital
Digital (77, 86, large)-net over F9, using
- t-expansion [i] based on digital (76, 86, large)-net over F9, using
- 3 times m-reduction [i] based on digital (76, 89, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(989, large, F9, 13) (dual of [large, large−89, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(989, large, F9, 13) (dual of [large, large−89, 14]-code), using
- 3 times m-reduction [i] based on digital (76, 89, large)-net over F9, using
(77, 77+9, large)-Net in Base 9 — Upper bound on s
There is no (77, 86, large)-net in base 9, because
- 7 times m-reduction [i] would yield (77, 79, large)-net in base 9, but