Best Known (77, 86, s)-Nets in Base 8
(77, 86, 4456447)-Net over F8 — Constructive and digital
Digital (77, 86, 4456447)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 20, 262147)-net over F8, using
- net defined by OOA [i] based on linear OOA(820, 262147, F8, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(820, 262147, F8, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(820, 524294, F8, 4) (dual of [524294, 524274, 5]-code), using
- trace code [i] based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(647, 262144, F64, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 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(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(820, 524294, F8, 4) (dual of [524294, 524274, 5]-code), using
- appending kth column [i] based on linear OOA(820, 262147, F8, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(820, 262147, F8, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
- digital (57, 66, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(866, 8388602, F8, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- trace code [i] based on linear OOA(6433, 4194301, F64, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- trace code [i] based on linear OOA(6433, 4194301, F64, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(866, 8388602, F8, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- digital (16, 20, 262147)-net over F8, using
(77, 86, large)-Net over F8 — Digital
Digital (77, 86, large)-net over F8, using
- t-expansion [i] based on digital (76, 86, large)-net over F8, using
- 3 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, 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(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- 3 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
(77, 86, large)-Net in Base 8 — Upper bound on s
There is no (77, 86, large)-net in base 8, because
- 7 times m-reduction [i] would yield (77, 79, large)-net in base 8, but