Best Known (69, 69+9, s)-Nets in Base 8
(69, 69+9, 4198332)-Net over F8 — Constructive and digital
Digital (69, 78, 4198332)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 4032)-net over F8, using
- net defined by OOA [i] based on linear OOA(812, 4032, F8, 4, 4) (dual of [(4032, 4), 16116, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(812, 4032, F8, 3, 4) (dual of [(4032, 3), 12084, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(812, 8064, F8, 4) (dual of [8064, 8052, 5]-code), using
- trace code [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- 1 times truncation [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- trace code [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(812, 8064, F8, 4) (dual of [8064, 8052, 5]-code), using
- appending kth column [i] based on linear OOA(812, 4032, F8, 3, 4) (dual of [(4032, 3), 12084, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(812, 4032, F8, 4, 4) (dual of [(4032, 4), 16116, 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 (8, 12, 4032)-net over F8, using
(69, 69+9, large)-Net over F8 — Digital
Digital (69, 78, large)-net over F8, using
- 3 times m-reduction [i] based on digital (69, 81, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, 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(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
(69, 69+9, large)-Net in Base 8 — Upper bound on s
There is no (69, 78, large)-net in base 8, because
- 7 times m-reduction [i] would yield (69, 71, large)-net in base 8, but