Best Known (64−8, 64, s)-Nets in Base 8
(64−8, 64, 4194328)-Net over F8 — Constructive and digital
Digital (56, 64, 4194328)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 28)-net over F8, using
- net defined by OOA [i] based on linear OOA(86, 28, F8, 4, 4) (dual of [(28, 4), 106, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(86, 28, F8, 3, 4) (dual of [(28, 3), 78, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- appending kth column [i] based on linear OOA(86, 28, F8, 3, 4) (dual of [(28, 3), 78, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(86, 28, F8, 4, 4) (dual of [(28, 4), 106, 5]-NRT-code), using
- digital (50, 58, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- digital (2, 6, 28)-net over F8, using
(64−8, 64, large)-Net over F8 — Digital
Digital (56, 64, large)-net over F8, using
- t-expansion [i] based on digital (55, 64, large)-net over F8, using
- 1 times m-reduction [i] based on digital (55, 65, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- 1 times m-reduction [i] based on digital (55, 65, large)-net over F8, using
(64−8, 64, large)-Net in Base 8 — Upper bound on s
There is no (56, 64, large)-net in base 8, because
- 6 times m-reduction [i] would yield (56, 58, large)-net in base 8, but